Derivations and central extensions of simple modular Lie algebras and superalgebras
Sofiane Bouarroudj${}^1$
Pavel Grozman${}^2$
Alexei Lebedev${}^2$
Dimitry Leites${}^3$
Over algebraically closed fields of positive characteristic, we list the outer derivations and non-trivial central extensions of simple finite dimensional Lie algebras and Lie superalgebras of the rank sufficiently small to allow computer-aided study using Mathema\-tica-based code SuperLie. We consider algebras possessing a grading by integers such that the dimensions of the superspaces whose degrees are equal in absolute value coincide (symmetric algebras), and also non-trivial deforms of these algebras. When the answer is clear for the infinite series, it is given for any rank. We also consider the non-symmetric case of periplectic Lie superalgebras and the Lie algebras obtained from them by desuperization.
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\begin{document}

\title{Derivations and central extensions of simple modular Lie algebras and
superalgebras}


\author{Sofiane Bouarroudj${}^1$, Pavel Grozman${}^2$, Alexei Lebedev${}^2$,
Dimitry Leites${}^3$}

\address{${}^1$New York University Abu Dhabi,
Division of Science and Mathematics, P.O. Box 129188, United Arab
Emirates; sofiane.bouarroudj@nyu.edu \\
${}^2$Equa
Simulation AB, Stockholm, Sweden; pavel@rixtele.com; yorool@mail.ru\\
${}^3$Department of Mathematics, Stockholm University, Roslagsv.
101, Kr\"aft\-riket hus 6, SE-106 91 Stockholm, Sweden;
mleites@math.su.se}

\keywords{modular Lie superalgebra}  \subjclass{17B50, 70F25}

\begin{abstract} Over algebraically closed fields of
positive characteristic, we list the outer derivations and
non-trivial central extensions of simple finite dimensional Lie
algebras and Lie superalgebras of the rank sufficiently small to
allow computer-aided study using Mathema\-tica-based code SuperLie.
We consider algebras possessing a grading by integers such that the
dimensions of the superspaces whose degrees are equal in absolute
value coincide (symmetric algebras), and also non-trivial deforms of
these algebras. When the answer is clear for the infinite series, it
is given for any rank.

We also consider the non-symmetric case of periplectic Lie
superalgebras and the Lie algebras obtained from them by
desuperization.
\end{abstract}

\thanks{S.B.
was partly supported by the grant AD 065 NYUAD}

%\date{Received May 1, 2006}

\maketitle

\markboth{\itshape Sofiane Bouarroudj\textup{,} Pavel
Grozman\textup{,} Alexei Lebedev\textup{,} Dimitry Leites}{{\itshape
Derivations and central extensions of simple Lie algebras}}


\thispagestyle{empty}


\section{Introduction}\texttt{}

Hereafter, $\Kee$ is an algebraically closed field of characteristic
$p>0$ and $\fg$ is a finite dimensional Lie (super)algebra.

\ssec{Goal: classify simple modular Lie algebras} This major problem
was attacked from the two directions.

\sssec{All algebras, but $p>5$} In late 1960s, Kostrikin and
Shafarevich formulated a conjecture describing a method producing
all simple finite dimensional restricted modular Lie algebras over
algebraically closed fields of characteristic $p>7$, and a
generalization embracing non-restricted algebras. The statement of
the conjecture turned out to hold for $p=7$ as well. The
Kostrikin-Shafarevich procedure for obtaining all simple finite
dimensional Lie algebras consists of the following steps:
\begin{equation}\label{Kan}
\begin{minipage}[l]{14cm}
\hskip4 mm 1) considering the two types of simple complex Lie
algebras:

a) those of the form $\fg(A)$ for a Cartan matrix $A$,

b) infinite dimensional vectorial Lie algebras with polynomial
coefficients;

\hskip 4 mm 2) selecting certain bases allowing integer structure
constants (Chevalley bases for the algebras of the form $\fg(A)$ and
divided powers for vectorial Lie algebras) thus getting $\Zee$-forms
of these complex Lie algebras;

\hskip 4 mm 3) tensoring the $\Zee$-forms obtained at step 2) by
$\Kee$ over $\Zee$;

\hskip 4 mm 4) selecting a simple (and finite dimensional in the
vectorial Lie algebra case) subquotient, called a simple ``relative"
in what follows;

\hskip 4 mm 5) deforming the results obtained at step 4);

\hskip 4 mm 6) classification of isomorphisms between Lie algebras
obtained at earlier steps.
\end{minipage}
\end{equation}

At first, addressing this classification problem (goal),
Shafarevich, together with his student, Kostrikin, only considered
{\it restricted} Lie algebras. This self-restriction was, perhaps,
occasioned by the fact that only restricted Lie algebras correspond
to algebraic groups, and being a geometer, albeit an algebraic one,
Shafarevich did not see much reason to consider non-restricted Lie
algebras. Recently, on a different occasion, Deligne gave us an
advice \cite{LL} which we interpret as indication to look at the
groups (geometry) rather than at Lie algebras if $p>0$. Since only
restricted Lie algebras correspond to groups we interpret Deligne's
advice as a certain natural indication to restrict the
classification problem of simple Lie (super)algebras to restricted
ones at the first step.

\sssec{All algebras, $p=5$} For $p=5$, Melikyan found a series of
algebras not produced by the method suggested by Kostrikin and
Shafarevich. More examples not fitting the KSh-method were known for
$p=3$ and 2.

Summing up 30 years of work of several teams of researchers, Block
and Wilson, Premet and Strade not only proved the KSh-conjecture,
but even completed the classification of all simple finite
dimensional Lie algebra over algebraically closed fields of
characteristic $p>3$; for reviews, see \cite{S, BGP}.

\sssec{Arbitrary $p$, but algebras only with indecomposable Cartan
matrix, or simple subquotient thereof} For a particular case of
finite dimensional Lie algebras $\fg(A)$ with indecomposable Cartan
matrix $A$ for any $p>0$, Weisfeiler and Kac \cite{WK} offered a
classification.

Although the idea of the proof in \cite{WK} is OK, the paper has
several gaps and vague notions: the Brown algebra $\fbr(3)$, see
\cite{Br3} was missed; the notion of the Lie algebra with Cartan
matrix nicely defined in \cite{K} was not properly developed at the
time \cite{WK} was written; the quotients $\fg(A)^{(i)}/\fc$ of the
first or second (for $i=1$ or 2) derived modulo center $\fc$ have no
Cartan matrix but were often referred as having one. All the needed
preliminary notions are carefully defined  in \cite{BGL1} where the
result of \cite{WK} with Skryabin's correction (he observed that
$\fbr(3)$ possesses a Cartan matrix, actually, two inequivalent
ones) is double-checked.

\sssec{A conjectural method for producing all simple Lie algebras
for $p\geq 3$} For its exposition, see \cite{Ltow}.

{\bf The main procedure} for constructing simple Lie (super)algebras
in the conjecture \cite{Ltow}, that claims to embrace $p=2$ as well,
is the \emph{generalized Cartan prolongation}.

{\bf The main characters} in this procedure are simple Lie
(super)algebras, their central extensions and algebras of
derivations, and modules over such algebras.

The process is inductive: Given a simple Lie (super)algebra, we
describe its ``relatives", i.e., central extensions and algebra of
derivations; they might be building bricks in the construction of a
new simple algebra. In various problems these relatives of simple
Lie (super)algebra are no less important than simple algebras.



\sssec{Simple algebras and their relatives with important
properties. Outer derivations and central extensions} For the Lie
(super)algebras with an (even) supersymmetric invariant bilinear
form $b$, the problems of description of these two objects (outer
derivations and non-trivial central extensions) are almost
equivalent. Indeed, the outer derivations are described by cocycles
representing the classes of $H^1(\fg; \fg)$ whereas the non-trivial
central extensions are defined by cocycles representing the classes
of $H^2(\fg)$, i.e., with trivial coefficients. In presence of such
a form $b$, we have $\fg\simeq\fg^*$, and hence $H^1(\fg; \fg)\simeq
H^1(\fg; \fg^*)$.

There are maps $i:H^1(\fg; \fg^*)\tto
H^2(\fg)$ and $j:H^2(\fg)\tto H^1(\fg; \fg^*)$ given, for any cocycle $f$ representing a class $[f]\in
H^1(\fg; \fg^*)$, any cocycle $\omega$ representing a class $[\omega]\in
H^2(\fg)$, and $X,Y\in \fg$, by the formulas
\begin{equation}\label{embh1toh2}
\begin{array}{l}
i(f)(X, Y)=f(X)(Y),\\
j(\omega)(X)(Y)=\omega(X, Y).
\end{array}
\end{equation}
The relation between these cohomology spaces is given by the following exact sequence
\begin{equation}\label{exact}
0 \tto H^2(\fg) \stackrel{j}{\tto} H^1(\fg; \fg^*)
\stackrel{v}{\tto} (S^2(\fg))^\fg \stackrel{K}{\tto} H^3(\fg)\tto
H^2(\fg; \fg^*) \tto H^1(\fg;S^2(\fg)),
\end{equation}
where $(S^2(\fg))^\fg$ denotes the space of symmetric bilinear $\fg$-invariant forms on $\fg$,
and the maps $v$ and $K$ are
defined as follows:
\[
\begin{array}{ll}
v(d)(X, Y) = d(X)(Y) + (-1)^{p(X)p(X)}d(Y)(X)&\text{for any $X,Y\in \fg$ and $d\in
Z^1(\fg; \fg^*)$},\\
K(b)(X, Y, Z) = b([X, Y], Z) &\text{for any $X,Y, Z\in \fg$ and $b\in
(S^2(\fg))^\fg$}.
\end{array}
\]
The beginning of the exact sequence \eqref{exact} was discovered by
Koszul, who introduced the map $K$, the exact sequence was further
extended to the right on different occasions, see \cite{NW, DzZ}.
The fact that the map $j$ is inclusion is proved in \cite{Dz5}; the
exactness of the sequence \eqref{exact} is proved in \cite{NW},
Proposition 7.2. For the case of \emph{even} invariant symmetric
bilinear forms $(S^2(\fg))^\fg$, the arguments of \cite{NW} are
literally applicable to Lie {\bf super}algebras.

``What is the relation between $H^2(\fg)$ and $H^1(\fg; \fg)$ if
$\fg$ possesses an \emph{odd} symmetric $\fg$-invariant bilinear
form?" is an {\bf open question}. Anyway, it is clear that if
$(S^2(\fg))^\fg=0$, there is an isomorphism $H^2(\fg)\simeq H^1(\fg;
\fg^*)$; this is wonderful, except that we are interested in
$H^1(\fg; \fg)$, instead. So we have to compute this space, anyway.
Or, the other way round, information about outer derivations does
not help to say what are the central extensions, and we have to
compute $H^2(\fg)$, anyway. There is, nevertheless, something we can
say a priori (and double-check the results).

\sssec{What can be said a priori} Let $\fg=\fg(A)$ be a Lie
(super)algebra with a degenerate Cartan matrix. Then $\fg$ has a
central element $C$ and a grading element $D$, both lying in the
maximal torus $\fh$ and $D\not\in\fg'$. Assume, for simplicity, that
$\rk(A)=\size(A)-1$, so $\fg = \fg'\subplus \Kee\cdot D$, a
semidirect sum with $\fg'$ an ideal.


\parbegin{Statement} The operator $M$ on $\fg$ corresponding to the cochain $C\otimes
d(D)$, i.e., such that
\begin{equation}\label{AL} MD=C;\quad M\fg'=0, \end{equation}
is an outer derivation of $\fg$.\end{Statement}

\begin{proof} To show that it is a derivation, we need to show that
\[
M[x,y]=[Mx,y]+[x,My] \ \text{for any $x,y\in\fg$}. \] But the
left-hand side is 0 thanks to \eqref{AL}, and the right-hand side is
0 due to the fact that $\IM M\subset \fc$, where $\fc$ is the center
of $\fg$.

This derivation is an outer one, i.e., there is no $x\in\fg$ such
that $M=\ad_x$. Indeed, otherwise $[x,D]=C$, but $D\in\fh$, so
$[D,\fh]=0$; $[D,\fg_\pm]\subset\fg_\pm$. Thus, $C\not\in [D,\fg]$
since $C\in\fh$. \end{proof}

If $\size(A)-\rk(A)=k$, then there are $k$ linearly independent
central elements $C_1,\dots,C_k$, and $k$ grading elements
$D_1,\dots,D_k$. Similarly to the above, there are $k^2$ linearly
independent outer derivations of $\fg$, i.e., operators $M_{ij}$
corresponding to the cochains $C_i\otimes d(D_j)$.



\sssec{When $p=2$ problems and obstacles pile up. Hidden
supersymmetries} It was clear since mid-1970s that to classify
simple Lie algebras in characteristic 2, we have to incorporate Lie
superalgebras: forgetting superstructure (``desuperizing") makes any
Lie superalgebra into a Lie algebra; this process often preserves
simplicity.

The other way round, ``hidden supersymmetries" of Lie algebras,
i.e., possibility to obtain Lie superalgebra with the same ``family
name" as a given Lie algebra, e.g., $\fgl$ or $\fe$, for ANY (both
simple and ``relative" thereof) Lie algebra.

Further cogitations on the case of $p=2$ and elucidations of the
basics made it clear (to A.~Lebedev, this was a Chapter in his Ph.D.
thesis, \cite{LeD}) that the existence of a 2-structure on a simple
Lie algebra leads to a new type of simple Lie superalgebras, the
type never noticed before: \emph{queerifications of restricted
simple Lie algebras}.

Since the lack of outer derivations is a sufficient condition for
the the algebra to be restricted, it was natural to describe these
derivations for the cases not considered in \cite{FG}, and --- since
this description is of independent interest
--- for Lie superalgebras. This is what we do in what follows
for all ``symmetric'' Lie (super)algebras of small rank, and in
several general cases (with high probability for all cases in these
dimensions, not just for the ``known" ones, thanks to the results by
Eick \cite{Ei}).

For simple $\Zee$-graded vectorial Lie algebras, non-trivial central extensions
are described mainly by A.~Dzhumadildaev, see \cite{Dz4,DzU}.

For ``lopsided'' Lie algebras not considered by Dzhumadildaev, and
``lopsided'' Lie superalgebras, (we have in mind only simple ones
and ``relatives" thereof), to compute the outer derivatives and
non-trivial central extensions is an {\bf open problem}.



\ssec{Super goal: classify simple modular Lie superalgebras}



\sssec{Lie superalgebras with indecomposable Cartan matrices, their
simple subquotients} These are classified in \cite{BGL1}. If $p=2$,
there is one more type of symmetric Lie algebras, the orthogonal Lie
algebras. Among the derived of the orthogonal Lie algebras, there is
also an important version of ``lopsided" Lie algebras, an analog of
the periplectic Lie superalgebras. One more type of the symmetric
Lie superalegbras is produced by restricted simple Lie algebras of
the form $\fg(A)$ or their subquotients.

The deforms of the symmetric Lie (super)algebras are studied in
\cite{BGL2}, results are partial.

All other types of simple Lie superalgebras are not classified yet.
There are only available conjectural methods for producing all of
them, see \cite{Ltow}.

\sssec{Simple vectorial Lie superalgebras} For $p>5$, one can,
conjecturally, use same Kostrikin-Shafarevich procedure \eqref{Kan}
with the following simple Lie superalgebras over $\Cee$ for the
input: of the form $\fg(A)$ (classified for various types of growth
in \cite{Ser, HS}, see also \cite{CCLL}); simple vectorial with
polynomial coefficients (classified in \cite{LSh}), together with
the special periplectic $\fspe(n)$ and ``queer" Lie superalgebras
$\fpsq(n)$ (first discovered in \cite{Dj}), and pass to simple
subquotients of all the above.

For $p=5$, the same conjecture is, perhaps, true; minus Melikyan
algebras and plus examples of the form $\fg(A)$ and related
subquotients, see \cite{BGL1, BGLL}.

For $p=3$, there appear new examples: For algebras, see \cite{GL4};
for superalgebras, see \cite{BL, BGL3}.

For $p=2$, there appear still new examples: For algebras, see
\cite{GL4}; for superalgebras, see \cite{LeP, BGLLSq}.


\section{Main results}
In this section we collected the description of outer derivations.

One of the main types or our examples are Lie (super)algebras with
Cartan matrix, or their ``relatives". Of course, the weights of
cocycles do not depend on a realization (the choice of Cartan
matrix) but the form of cocycles does, so for each Lie
(super)algebra, we give the result for one (``simplest" in some
sense) incarnation if there are several. We indicate the Cartan
matrix to which our cocycles correspond. Thanks to symmetry, it
suffices to give only cocycles of non-positive degree. We give only
the ``non-super" version of the Cartan matrix; for Lie superalgebras
with the ``same" Cartan matrix (but with 0 instead of $\ev$ on the
diagonal) the form of the cocycle is virtually the same, so we do
not list these super cases separately, bar several exceptional cases
where the answer is distinct (for reasons unknown).\DL{}

The answer for the simple subquotient $\fg^{(1)}(A)/\fc$ of the Lie
(super)algebra of the form $\fg(A)$ is given with respect to the
same basis $x$ for positive root vectors and $y$ for negative root
vectors as for $\fg(A)$ itself. The elements $h$ belong to the
maximal torus: $h_i:=[x_i,y_i]$; for the singular Cartan matrices,
we denote the elements $d_j$, see \cite{BGL1}, also by letters $h$
in order not to confuse them with the differential $d$. The
following cocycles are labeled by their degree defined by setting
$\deg x_i=-\deg y_i=1$ for every Chevalley generator (do not confuse
generators with the other elements of the Chevalley basis); the
superscript numerates cocycles of the same degree, if any such
occur.

Let $\fout\fg:=\faut(\fg)/\fg$ be the Lie (super)algebra of outer
derivations of $\fg$.

We consider algebras as rank grows, but if the general answer for a
series considered is clear, we give the answer at the first
instance.

 \ssec{Rank=1}

\sssec{$p=2$}

\parbegin{Lemma} For $\fo^{(1)}_{\Pi}(2n+1)$, the space $H^1(\fg;\fg)$
is spanned by $2n$ cocycles of weight $-2n$, $-2n+2$, \dots, $2n-2$,
$2n$ and $\faut\fg=\fo_{\Pi}(2n+1)$.
\end{Lemma}

\parbegin{Lemma} For $\fhei(2)$, we have $\fout\fg\simeq\fgl(2)$; and the
space $H^1(\fg;\fg)$ is spanned by the following cocycles:
\[
\begin{array}{ll}
c_{-2}=&y_1\otimes dx_1\\
c_{0}^1=&h_1\otimes dh_1+x_1\otimes dx_1\\
c_{0}^2=&x_1\otimes dx_1+y_1\otimes dy_1
\end{array}
\]
\end{Lemma}

\begin{Remark} The result is correct, though not symmetric.
Let us do it by hand. The basis of the space of 1-cochains of degree
0 is $\{h_1\otimes dh_1, \ x_1\otimes dx_1, \ y_1\otimes dy_1\}$. If
we compute the differential of a linear combination of these
cocycles with coefficients $i_s$ for $s=1, 2, 3$, we get
\begin{equation}\label{comb}
(i_1+i_2+i_3) h_1\otimes dx_1\wedge dy_1\  .
\end{equation}
For the linear combination to be a cocycle, the expression
\eqref{comb} should vanish, i.e., $i_1+i_2+i_3=0$. Besides, there
are now coboundaries of degree 0 because, $d(h_1)=0$. Now,
\emph{SuperLie} is taking the basis $i_1=i_2=1, i_3=0$ and $i_1=0,
i_2=i_3=1$. To have a symmetric answer, we'd take the basis with
$i_1=i_3=1, i_2=0$ and $i_1=i_2=1, i_3=0$; the code \emph{SuperLie}
has different esthetic criteria. We encounter the same type of
answer for $p=3$. \end{Remark}


\parbegin{Lemma} For $\fhei(0|2)$, we have $\fout\fg\simeq\fo_\Pi(2)\oplus\Kee z$;
the space $H^1(\fg;\fg)$ is spanned
by the following cocycles for $p=2$, $3$, $5$, \dots, and $0$:

\[
\begin{array}{l}
c_{0}^1=h_1\otimes dh_1-x_1\otimes dx_1\\
c_{0}^2=-x_1\otimes dx_1+y_1\otimes dy_1
\end{array}
\]
\end{Lemma}

\parbegin{Lemma}\label{oo1/2} For $\fo\fo_{I\Pi}(1|2)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\begin{Remark}One might expect the same result as for
$\fo^{(1)}_{\Pi}(3)$. Lemma \ref{oo1/2} is, however, correct: let us
do it by hand again. There is one coboundary: $x_1\otimes
dx_1+y_1\otimes dy_1$. The space of cochains is 5-dimensional, that
of cocycles is equal to 2. Now for $\fo^{(1)}_{\Pi}(3)$, the
situation is different (first, the dimension is smaller), the space
of cochains is 3-dimensional, and that of cocycles is equal to 1.
The coboundary, which is also $x_1\otimes dx_1+y_1\otimes dy_1$,
reduces to 0. We encounter the same type of answer for $p=3$.
\end{Remark}


\sssec{$p=3$}


\parbegin{Lemma} For $\fhei(2)$, we have $\fout\fg\simeq\fgl(2)$;
the space $H^1(\fg;\fg)$ is spanned
by the following cocycles
\[
\begin{array}{ll}
c_{-2}=&y_1\otimes dx_1\\
c_{0}^1=&h_1\otimes dh_1+\  x_1\otimes dx_1\\
c_{0}^2=&-x_1\otimes dx_1+y_1\otimes dy_1
\end{array}
\]
\end{Lemma}

\parbegin{Lemma} For $\fosp(1|2)$, the space $H^1(\fg;\fg)$ is
spanned by the following cocycles (here $x_2=[x_1,x_1]$ and the
$y_2$ is similarly defined):
\[
\begin{array}{ll}
c_{-3}=& 2 \ y_1\otimes d x_2+y_2\otimes dx_1
\end{array}
\]
\end{Lemma}

\parbegin{Lemma} For $\fsl(2)$, we have $H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p=5$}

\parbegin{Lemma} For $\fhei(2)$, we have $\fout\fg\simeq\fgl(2)$;
the space $H^1(\fg;\fg)$ is spanned
by the following cocycles:
\[
\begin{array}{ll}
c_{-2}=&y_1\otimes dx_1\\
c_{0}^1=&h_1\otimes dh_1-x_1\otimes dx_1\\
c_{0}^2=&-x_1\otimes dx_1+y_1\otimes dy_1
\end{array}
\]
\end{Lemma}

\parbegin{Lemma} For $\fg=\fsl(2)$ and $\fosp(1|2)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\ssec{Rank=2}

\sssec{$p=2$}

%\parbegin{Lemma} For $\fo^{(1)}_{\Pi}(5)$ with Cartan matrix
%\begin{equation}
%\begin{pmatrix}
% \ev& 1 \\
% 1 & \od\end{pmatrix}\quad\text{and the basis } \;
%\begin{array}{l}
%x_1,\ x_2,\\
%x_3=[x_1, x_2],\\
%x_4=[x_1, [x_1, x_2]],
%\end{array}\end{equation} The $H^1(\fg;\fg)$ is spanned by the following cocycles:
%\[
%\begin{array}{ll}
%c_{-4}=& y_2\otimes dx_4+y_3\otimes dx_3
%+y_4\otimes dx_2\\
%c_{ -2}=& x_2\otimes dx_4+y_1\otimes dx_1
% +y_4\otimes dy_2
%\end{array}
%\]
%\end{Lemma}

\parbegin{Lemma} For $\fo \fo_{I \Pi}^{(1)}(3|2)$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 0& 1 \\
 1 & 1 \end{pmatrix}\quad\text{and the basis } \;
\begin{array}{l}
x_1,\ x_2,\\
x_3=x_2^2,x_4=[x_1, x_2],\\
x_5=[x_2, x_2^2],
\end{array}\end{equation}
the space $H^1(\fg;\fg)$ is spanned by the following cocycles:
\[
\begin{array}{ll}
c_{-4}=&y_2\otimes dx_5+y_4\otimes dx_4 +y_5\otimes dx_2
\end{array}
\]
\end{Lemma}

\parbegin{Lemma} For $\fg=\fsl(3)$, $\fsl(1|2)$, $\fo\fo_{I
\Pi}^{(1)}(1|2)$, $\fo\fo_{I \Pi}^{(1)}(1|4)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p=3$} \parbegin{Lemma} a) For $\fg=\fgl(3)$, the space
$H^1(\fg;\fg)$ is spanned by the following cocycle:
\[
c_{0}=2\  h_1\otimes dh_3
 +h_2\otimes dh_3
 \]
Above $h_3$ denotes the grading operator in order not to confuse it
with the differential $d$.

b) For $\fg=\fpsl(3)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycles:
\[
{\footnotesize\begin{array}{ll}
c_{-3}=& y_1\otimes d x_3+x_3\otimes d x_1+y_2\otimes d x_3+x_3\otimes d x_2\\[3mm]

c_{0}^1=& 2\  x_2\otimes d x_1+y_1\otimes d y_2\\

c_{0}^2=& x_1\otimes d x_1 +2\  x_2\otimes d x_2 +2\
y_1\otimes d y_1 +y_2\otimes d y_2\\

c_{0}^3=& 2\  x_1\otimes d x_2+y_2\otimes d y_1
\end{array}
} \]
\end{Lemma}

\begin{Remark} This remark is for those who can not believe the above answer in heading
a) since ``$\fgl$ can not have outer derivations by definition". In
a way, the expression in quotation marks is not true if $p\neq 0$.
Indeed, the Cartan matrix of $\fgl(3)$ for $p=3$ is not invertible,
so there is a grading operator $d$, see \cite{BGL1}. For such a $d$
one can take, i.e., $h_3=2 E_{1,1}+E_{2,2}+E_{3,3}$, which, of
course, belongs to the algebra.\end{Remark}


\parbegin{Lemma} For $\fg=\fbrj(2;3)$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 0& -1 \\
 -2 & 1 \end{pmatrix}\quad\text{and the basis }{\footnotesize
 \;\begin{array}{l}
x_1,\;\;x_2,\\
x_3=[x_1, x_2],\ x_4=[x_2, x_2],\\
x_5=[x_2,
[x_1, x_2]],\\
x_6=[[x_1, x_2], [x_2, x_2]],\\
x_7=[[x_2, x_2], [x_2, [x_1, x_2]]],\\
x_8=[[x_2, [x_1, x_2]], [x_2, [x_1, x_2]]],\end{array}
}\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycles:
\[
c_{-3}= x_1\otimes dx_6+x_3\otimes dx_7+2\  y_2\otimes
dx_4+y_4\otimes dx_2+2\  y_6\otimes dy_1+y_7\otimes dy_3
\]
\end{Lemma}



\parbegin{Lemma} For $\fg=\fbr(3)$, $\fsl(1|2)$, $\fo(5)$,
$\fosp(3|2)$, $\fosp(1|4)$, we have $H^1(\fg;\fg)=0$.
\end{Lemma}


\sssec{$p=5$}

\parbegin{Lemma} For $\fg=\fbrj (2;5)$, $\fsl(3)$, $\fo(5)$,
$\fsl(1|2)$, $\fosp(1|2)$, and $\fosp(1|4)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}



\ssec{Rank=3}

\sssec{$p=2$}

\parbegin{Lemma} a) For $\fg=\fgl(4)$,
the space $H^1(\fg;\fg)$ is spanned by the following cocycle (here
$h_4$ is the outer derivation; same situation as for $\fgl(3)$ for
$p=3$):
\[
c_{0}=h_1\otimes dh_4+h_3\otimes dh_4
\]

b) For $\fg=\fpsl(4)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycles and the corresponding subalgebra of $\faut\fg$ is
indicated on the right if we can identify it:
\[{\footnotesize
\begin{array}{ll|l}
c_{-4}=& y_2\otimes dx_6+y_4\otimes d x_5+y_5\otimes d x_4+y_6
\otimes d
x_2&{\bf F}(\fh(0|4))\\[3mm]

c_{-2}=& x_2\otimes d x_6+y_1\otimes d x_3+y_3\otimes d
x_1+y_6\otimes d
y_2\\[3mm]

c_{0}^1=&x_3\otimes d x_1 +x_5\otimes d x_4
 +y_1\otimes d y_3 +y_4\otimes d y_5\\

c_{0}^2=&x_1\otimes d x_3 +x_4\otimes d x_5
 +y_3\otimes d y_1 +y_5\otimes d y_4\\

c_{0}^3=&x_2\otimes d x_2 +x_3\otimes d x_3
 +x_4\otimes d x_4 +y_2\otimes d y_2 +&\\
&y_3\otimes d y_3
 +y_4\otimes d y_4&\fpgl(4)\simeq {\bf
F}(\fh'(0|4)\subplus \Kee E)
 \end{array}
} \]

The non-trivial central extensions of $\fpsl(4)$: the right-most
column contains the description of the result when we can identify
the algebra; the independent cocycles are as follows:
\begin{equation}\label{centextrpsl4}
\begin{array}{ll|l}
c_{-4}=& d x_2 \wedge d x_6 + d x_4 \wedge d x_5&{\bf F}(\fpo'(0|4)) \\
c_{-2}=& d x_1 \wedge d x_3 + d x_6\wedge dy_2\\
c_{0}^1=& d x_1 \wedge d y_3 + d x_4 \wedge d y_5 \\
c_{0}^2=& d x_3 \wedge d y_1 + d x_5 \wedge d y_4 \\
c_{0}^3=& d x_2 \wedge d y_2 + d x_3 \wedge d y_3
 + d x_4\wedge d y_4&\fsl(4)
\end{array}
\end{equation}
\end{Lemma}




%\parbegin{Lemma} For $\fg=\fo^{(1)}_\Pi(7)$ with Cartan matrix
%\begin{equation}
%\begin{pmatrix}
% \ev & 1 & 0 \\
% 1 & \ev & 1 \\
% 0 & 1 & \od \end{pmatrix} \text{ with basis }{\footnotesize
% \begin{array}{l}
%x_1,\ x_2,\ x_3,\\
%x_4=[x_1, x_2],\ x_5=[x_2, x_3],\\
%x_6=[x_3,[x_1, x_2]],\ x_7=[x_3, [x_2, x_3]],\\
%x_8=[x_3,[x_3, [x_1, x_2]]],\\
%x_9=[[x_2, x_3],[x_3, [x_1, x_2]]]\end{array} }
%\end{equation}
%the space $H^1(\fg;\fg)$ is spanned by the following cocycles:
%\[
%\begin{array}{ll}
%c_{-6}=& y_1\otimes dx_9 +y_4\otimes dx_8 +y_6\otimes dx_6
%+y_8\otimes dx_4 +y_9\otimes dx_1 \\%[3mm]

%c_{-4}=& x_1\otimes dx_9 +y_2\otimes dx_7 +y_5\otimes dx_5
%+y_7\otimes dx_2 +y_9\otimes dy_1 \\%[3mm]

%c_{-2}=& x_2\otimes dx_7 +x_4\otimes dx_8 +y_3\otimes dx_3
%+y_7\otimes dy_2 +y_8\otimes dy_4 \\
% \end{array}
% \]
%\end{Lemma}

\parbegin{Lemma} a) For $\fg=\fwk(3,\alpha)$ and $\fg=\fbgl(3,\alpha)$ with Cartan matrix
\begin{equation}\label{wk3a}
\begin{pmatrix}
 \ev & \alpha & 0 \\
 \alpha & \ev & 1 \\
 0 & 1 & \ev\end{pmatrix}\;\text{ and basis }{\footnotesize
 \begin{array}{l}
 x_1,\;\;x_2,\;\;x_3,\\
 x_4=[x_1, x_2],\;\;x_5=[x_2, x_3],\\
 x_6=[x_3\text{,$\
 $}[x_1, x_2]],\\
 x_7=[[x_1, x_2], [x_2\text{,$\
 $}x_3]]\end{array}
}
\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycle:
\[
c_{0}=h_1\otimes dh_4+\alpha \
 h_3\otimes dh_4
 \]


b) For $\fg=\fwk^{(1)}(3,a)/\fc$ and $\fg=\fbgl^{(1)}(3,a)/\fc$, the
space $H^1(\fg;\fg)$ is spanned by the following cocycle:
\[
c_{0}=x_1\otimes d x_1+x_3\otimes d x_3 +x_4\otimes d x_4+x_5\otimes
d x_5 +y_1 \otimes d y_1+y_3\otimes d y_3+y_4 \otimes d y_4+
y_5\otimes d y_5
\]
\end{Lemma}




\sssec{$p=3$}

\parbegin{Lemma} a) For $\fg=\fgl(2|2)$, the space $H^1(\fg;\fg)$ is
spanned by the following cocycle(here $h_4$ is the outer derivation;
same situation as for $\fgl(3)$ for $p=3$):
\[
c_{0}=2\  h_1\otimes dh_4 + h_2\otimes dh_4 + h_3\otimes dh_4
\]

b) For $\fg=\fpsl(2|2)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycle:
\[
c_{0}=2 x_3 dx_3 + 2 x_4 dx_4 +2 x_5 dx_5 + y_3 dy_3 + y_4 dy_4 +
y_5 dy_5, \ \faut\fg=\fpgl(2|2).
 \]
\end{Lemma}


\parbegin{Lemma} a) For $\fg=\fg(2,3)$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 2 & -1 & -1 \\
 -1 & 2 & -1 \\
 -1 & -1 & 0\end{pmatrix} \text{ and basis }{\footnotesize
 \begin{array}{l}
x_1,\;\;x_2,\;\;x_3,\\
x_4=[x_1, x_2],\;\;
x_5=[x_1, x_3],\;\;x_6=[x_2, x_3],\\
x_7=[x_3, [x_1, x_2]],\\
x_8=[[x_1, x_2], [x_1, x_3]],\;\; x_9=[[x_1,
x_2], [x_2, x_3]],\\
x_{10}=[[x_1, x_2], [x_3, [x_1, x_2]]],\\
x_{11}=[[x_3, [x_1, x_2]], [x_3, [x_1, x_2]]]\end{array}
}
\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycle:
\[
c_{0}=2\  h_1\otimes dh_4 + h_2\otimes dh_4
\]


b) For $\fg=\fbj:=\fg(2,3)^{(1)}/\fc$, the space $H^1(\fg;\fg)$ is
spanned by the following cocycles:
\[
\begin{array}{ll}
c_{-3}=& 2\
 x_3\otimes d x_8+x_6\otimes d x_{10}+y_1\otimes d x_4+
 y_4\otimes d x_1+2\ y_8\otimes d y_3+y_{10}\otimes d y_6\\

c_{-3}=&x_3\otimes d x_9+x_5\otimes d x_{10}+2\
 y_2\otimes d x_4+2\
 y_4\otimes d x_2+y_9\otimes d y_3+y_{10}\otimes d y_5\\[3mm]

c_{0}^1=& x_2\otimes d x_1+2\  x_6\otimes d x_5+2\  x_9\otimes d
x_8+2\  y_1\otimes d y_2+y_5\otimes d y_6+y_8\otimes d y_9\\

c_{0}^2=& x_1\otimes d x_1+2\  x_2\otimes d x_2+2\  x_5\otimes d
x_5+x_6\otimes d x_6+2\
x_8\otimes d x_8+x_9\otimes d x_9\\
&+2\ y_1\otimes d y_1+y_2\otimes d y_2+y_5\otimes d y_5+2\
y_6\otimes d y_6+y_8\otimes d y_8+2\
y_9\otimes d y_9\\

c_{0}^3=&x_1\otimes d x_2+2\  x_5\otimes d x_6+2\  x_8\otimes d
x_9+2\  y_2\otimes d y_1+y_6\otimes d y_5+y_9\otimes d y_8
 \end{array}
 \]
\end{Lemma}

\parbegin{Lemma} For $\fg=\fg(1,6), \fosp(1|6), \fosp(2|4),
\fosp(3|4), \fosp(5|2), \fsl(1|3), \fosp(4|2)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p=5$}

\parbegin{Lemma} For $\fg=\fag(2)$, $\fosp(1|6)$, $\fosp(2|4)$,
$\fosp(3|4)$, $\fosp(5|2)$, $\fsl(1|3)$, $\fosp(4|2)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\parbegin{Lemma} a) For $\fg=\fgl(2|2)$, the space $H^1(\fg;\fg)$ is spanned
by the following cocycle (see comment on $h_4$ above):
\[
c_{0}=4\  h_1\otimes dh_4 + 3\  h_2\otimes dh_4 + h_3\otimes dh_4
\]

b) For $\fg=\fpsl(2|2)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycle:
\[
c_{0}=4 x_3 d_3 + 4 x_4 dx_4 +4 x_5 dx_5 + y_3 dy_3 + y_4 dy_4 + y_5
dy_5, \ \faut\fg=\fpgl(2|2).
 \]
\end{Lemma}


\ssec{Rank=4}

\sssec{$p=2$}

\parbegin{Lemma} (b) For $\fg=\fo\fc(1;8)\subplus \Kee I_0$,
$\fo\fo\fc(1;4|4)\oplus \Kee I_0$ and $\fo\fo\fc(1;6|2)/\fc$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 \ev & 0 & 0 & 1 \\
 0 & \ev & 0 & 1 \\
 0 & 0 & \ev& 1 \\
 1 & 1 & 1 & \ev \end{pmatrix} \text{ and basis }{\footnotesize
\begin{array}{l}
x_1,\;\;x_2,\;\;x_3,\;\;x_4,\\
x_5=[x_1, x_4],\;\; x_6=[x_2, x_4],\;\;
x_7=[x_3, x_4],\\
x_8=[x_2, [x_1, x_4]],\;\;x_9=[x_3, [x_1, x_4]],\\
x_{10}=[x_3, [x_2, x_4]], x_{11}=[x_3, [x_2, [x_1, x_4]]],\\
x_{12}=[[x_3, x_4], [x_2, [x_1, x_4]]]
\end{array}
}
\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycles:
\[
\begin{array}{l}
c_{0}^1=h_1\otimes dh_5 + h_2\otimes dh_5 \\

c_{0}^2=h_1\otimes dh_6 + h_2\otimes dh_6 \\

c_{0}^3=h_1\otimes dh_5 + h_3\otimes dh_5 \\

c_{0}^4=h_1\otimes dh_6 + h_3\otimes dh_6
 \end{array}
\]

(b) For $\fg=\fo\fc(1;8)/\fc$, $\fo\fo\fc(1;4|4)/\fc$ and
$\fo\fo\fc(1;6|2)/\fc$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycles: \footnotesize
\[
\begin{array}{ll}
c_{-6}^1=&y_1\otimes d x_{12} +y_5\otimes d x_{11} +y_8\otimes d x_9
+y_9\otimes d x_8 +y_{11}\otimes d x_5
+y_{12}\otimes d x_1\\

c_{-6}^2=&y_2\otimes d x_{12} +y_6\otimes d x_{11} +y_8\otimes d
x_{10} +y_{10}\otimes d x_8 +y_{11}\otimes d x_6
+y_{12}\otimes d x_2\\

c_{-6}^3=&y_3\otimes d x_{12} +y_7\otimes d x_{11} +y_9\otimes d
x_{10} +y_{10}\otimes d x_9 +y_{11}\otimes d x_7
+y_{12}\otimes d x_3\\[3mm]

c_{-4}^1=&x_1\otimes d x_{12} +y_4\otimes d x_{10} +y_6\otimes d x_7
+y_7\otimes d x_6 +y_{10}\otimes d x_4
+y_{12}\otimes d y_1\\

c_{-4}^2=&x_2\otimes d x_{12} +y_4\otimes d x_9 +y_5\otimes d x_7
+y_7\otimes d x_5 +y_9\otimes d x_4
+y_{12}\otimes d y_2\\

c_{-4}^3=&x_3\otimes d x_{12} +y_4\otimes d x_8 +y_5\otimes d x_6
+y_6\otimes d x_5 +y_8\otimes d x_4
+y_{12}\otimes d y_3\\[3mm]

c_{-2}^1=&x_4\otimes d x_{10} +x_5\otimes d x_{11} +y_2\otimes d x_3
+y_3\otimes d x_2 +y_{10}\otimes d y_4
+y_{11}\otimes d y_5\\

c_{-2}^2=&x_4\otimes d x_9 +x_6\otimes d x_{11} +y_1\otimes d x_3
+y_3\otimes d x_1 +y_9\otimes d y_4
+y_{11}\otimes d y_6\\

c_{-2}^3=&x_4\otimes d x_8 +x_7\otimes d x_{11} +y_1\otimes d x_2
+y_2\otimes d x_1 +y_8\otimes d y_4
+y_{11}\otimes d y_7\\[3mm]

c_{0}^1=&x_3\otimes d x_2 +x_7\otimes d x_6 +x_9\otimes d x_8
+y_2\otimes d y_3
+y_6\otimes d y_7 +y_8\otimes d y_9\\

c_{0}^2=&x_3\otimes d x_1 +x_7\otimes d x_5 +x_{10}\otimes d x_8
+y_1\otimes d y_3
+y_5\otimes d y_7 +y_8\otimes d y_{10}\\

c_{0}^3=&x_2\otimes d x_3 +x_6\otimes d x_7 +x_8\otimes d x_9
+y_3\otimes d y_2
+y_7\otimes d y_6 +y_9\otimes \text{dy}_{8}\\

c_{0}^4=&x_2\otimes d x_1 +x_6\otimes d x_5 +x_{10}\otimes d x_9
+y_1\otimes d y_2
+y_5\otimes d y_6 +y_9\otimes d y_{10}\\

c_{0}^5=&x_1\otimes d x_3 +x_5\otimes d x_7 +x_8\otimes d x_{10}
+y_3\otimes d y_1
+y_7\otimes d y_5 +y_{10}\otimes d y_8\\

c_{0}^6=&x_1\otimes d x_2 +x_5\otimes d x_6 +x_9\otimes d x_{10}
+y_2\otimes d y_1
+y_6\otimes d y_5 +y_{10}\otimes d y_9\\

c_{0}^7=&x_1\otimes d x_1 +x_2\otimes d x_2 +x_5\otimes d x_5
+x_6\otimes d x_6 +
x_9\otimes d x_9 +x_{10}\otimes d x_{10}+\\

&y_1\otimes d y_1 +y_2\otimes d y_2 +y_5\otimes d y_5
+y_6\otimes d y_6 +y_9\otimes d y_9 +y_{10}\otimes d y_{10}\\

c_{0}^8=&x_1\otimes d x_1+x_3\otimes d x_3+x_5\otimes d
x_5+x_7\otimes d x_7+ x_8\otimes d x_8+x_{10}\otimes d
x_{10}+\\

&y_1\otimes d y_1+y_3\otimes d y_3+ y_5\otimes d y_5+y_7\otimes d
y_7+y_8\otimes d y_8+y_{10}\otimes d y_{10}
 \end{array}
 \]
 \normalsize
\end{Lemma}




%\parbegin{Lemma} For $\fg=\fo^{(1)}(9)$ with Cartan matrix
%\begin{equation}
%\begin{pmatrix}
% \ev& 1 & 0 & 0 \\
% 1 & \ev & 1 & 0 \\
% 0 & 1 & \ev & 1 \\
% 0 & 0 & 1 & \od \end{pmatrix} \text{ and basis }{\footnotesize
%\begin{array}{l}
%x_1,\ x_2,\ x_3,\ x_4,\\
%x_5=[x_1, x_2],\ x_6=[x_2, x_3],\ x_7=[x_3, x_4],\\
%x_8=[x_3, [x_1, x_2]],\ x_9=[x_4, [x_2, x_3]],\ x_{10}=[x_4, [x_3,
%x_4]],\\
%x_{11}=[x_4, [x_4, [x_2, x_3]]],\ x_{12}=[[x_1, x_2], [x_3,
%x_4]],\\
%x_{13}=[[x_1, x_2], [x_4, [x_3, x_4]]],x_{14}=[[x_3, x_4], [x_4,
%[x_2, x_3]]],\\
%x_{15}=[[x_3, [x_1, x_2]], [x_4, [x_3, x_4]]],\\
%x_{16}=[[x_4, [x_2, x_3]], [[x_1, x_2], [x_3, x_4]]]\end{array} }
%\end{equation}
%the space $H^1(\fg;\fg)$ is spanned by the following cocycles:
%\footnotesize
%\[
%\begin{array}{ll}
%c_{-8}=& y_1\otimes dx_{16} +y_5\otimes dx_{15} +y_8\otimes dx_{13}
%+y_{12}\otimes dx_{12} + y_{13}\otimes dx_8 +y_{15}\otimes d
%x_5 +y_{16}\otimes dx_1 \\%[3mm]

%c_{-6}=&x_1\otimes dx_{16} +y_2\otimes dx_{14} + y_6\otimes dx_{11}
%+y_9\otimes dx_9 + y_{11}\otimes dx_6 +y_{14}\otimes dx_2
%+y_{16}\otimes dy_1 \\%[3mm]

%c_{-4}=&x_2\otimes dx_{14} + x_5\otimes dx_{15} +y_3\otimes dx_{10}
%+y_7\otimes dx_7 +y_{10}\otimes dx_3 +y_{14}\otimes dy_2
%+y_{15}\otimes dy_5 \\%[3mm]

%c_{-2}=& x_3\otimes dx_{10} +x_6\otimes dx_{11} + x_8\otimes dx_{13}
%+y_4\otimes dx_4 + y_{10}\otimes dy_3 +y_{11}\otimes dy_6
%+y_{13}\otimes dy_8
%\end{array}
%\]
%\normalsize
%\end{Lemma}

\parbegin{Lemma} For $\fg=\fo\fo_{I\Pi}^{(1)}(3|6)$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 \ev & 1 & 0 & 0 \\
 1 & \ev & 1 & 0 \\
 0 & 1 & 0 & 1 \\
 0 & 0 & 1 & \od \end{pmatrix} \text{ and basis }{\footnotesize
\begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\\
x_5=[x_1, x_2],\ x_6=[x_2, x_3],\ x_7=[x_3, x_4],\\
x_8=[x_3, [x_1, x_2]],\ x_9=[x_4, [x_2, x_3]],\ x_{10}=[x_4, [x_3,
x_4]],\\
x_{11}=[x_4, [x_4, [x_2, x_3]]],\ x_{12}=[[x_1, x_2], [x_3,
x_4]],\\
x_{13}=[x_3, x_4]{}^2, \ x_{14}=[[x_1, x_2], [x_4, [x_3,
x_4]]],\\
x_{15}=[[x_3, x_4], [x_4, [x_2, x_3]]],\\
x_{16}=[[x_3, [x_1, x_2]], [x_4, [x_3, x_4]]],\\
x_{17}=[x_4, [x_2, x_3]]{}^2, \ x_{18}=[[x_4, [x_2, x_3]],
[[x_1, x_2], [x_3, x_4]]],\\
x_{19}=[[x_1, x_2], [x_3, x_4]]{}^2\end{array} }
\end{equation}
the space $H^1(\fg;\fg)$ is spanned by the following cocycles:
\footnotesize
\[
\begin{array}{ll}
c_{-2}=& x_3\otimes dx_{10} +x_6\otimes dx_{11} + x_8\otimes dx_{14}
+y_4\otimes dx_4 + y_{10}\otimes dy_3 +y_{11}\otimes dy_6
+y_{14}\otimes dy_8
\end{array}
\]
\normalsize
\end{Lemma}

\parbegin{Lemma} For $\fg=\fo\fo_{I\Pi}^{(1)}(5|4)$ with Cartan matrix
\begin{equation}
\begin{pmatrix}
 \ev & 1 & 0 & 0 \\
 1 & 0 & 1 & 0 \\
 0 & 1 & \ev& 1 \\
 0 & 0 & 1 & \od \end{pmatrix} \text{ and basis }{\footnotesize
 \begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\\
x_5=[x_1, x_2],\ x_6=[x_2, x_3],\ x_7=[x_3, x_4],\\
x_8=[x_3, [x_1, x_2]],\ x_9=[x_4, [x_2, x_3]],\ x_{10}=[x_4, [x_3,
x_4]],\\
x_{11}=[x_4, [x_4, [x_2, x_3]]],x\ _{12}=[[x_1, x_2], [x_3,
x_4]],\\
x_{13}=[[x_1, x_2], [x_4, [x_3, x_4]]],\ x_{14}=[[x_3, x_4], [x_4,
[x_2, x_3]]],\\
x_{15}=[[x_3, [x_1, x_2]], [x_4, [x_3, x_4]]],\\
x_{16}=[x_4, [x_2, x_3]]{}^2,\ x_{17}=[[x_4, [x_2, x_3]],
[[x_1, x_2], [x_3, x_4]]],\\
x_{18}=[[x_1, x_2], [x_3, x_4]]{}^2\end{array} }
\end{equation} the
space $H^1(\fg;\fg)$ is spanned by the following cocycles:
\footnotesize
\[
\begin{array}{ll}
c_{-4}=& x_2\otimes dx_{14} +x_5\otimes dx_{15} +y_3\otimes dx_{10}
+ y_7\otimes dx_7 +y_{10}\otimes dx_3 +y_{14}\otimes dy_2 +
y_{15}\otimes dy_5 \\
c_{-2}=& x_3\otimes dx_{10} +x_6\otimes dx_{11} +x_8\otimes dx_{13}
+y_4\otimes dx_4 +y_{10}\otimes dy_3 +y_{11}\otimes dy_6 +
y_{13}\otimes dy_8
\end{array}
\]
\normalsize

\end{Lemma}

\parbegin{Lemma} For $\fg=\fo\fo_{I\Pi}^{(1)}(7|2)$, with Cartan matrix
\begin{equation}
\begin{pmatrix}
 0 & 1 & 0 & 0 \\
 1 & \ev & 1 & 0 \\
 0 & 1 & \ev & 1 \\
 0 & 0 & 1 & 1\end{pmatrix} \text{ and basis }{\footnotesize
 \begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\\
x_5=[x_1, x_2],\ x_6=[x_2, x_3],\ x_7=[x_3, x_4],\\
x_8=[x_3, [x_1, x_2]],\ x_9=[x_4, [x_2, x_3]],x_{10}=[x_4, [x_3,
x_4]],\\
x_{11}=[x_4, [x_4, [x_2, x_3]]],\ x_{12}=[[x_1, x_2], [x_3,
x_4]],\\
x_{13}=[[x_1, x_2], [x_4, [x_3, x_4]]],\ x_{14}=[[x_3, x_4], [x_4,
[x_2, x_3]]],\\
x_{15}=[[x_3, [x_1, x_2]], [x_4, [x_3, x_4]]],\\
x_{16}=[[x_4, [x_2, x_3]], [[x_1, x_2], [x_3, x_4]]],\\
x_{17}=[[x_1, x_2], [x_3, x_4]]{}^2\end{array} }
\end{equation} The
space $H^1(\fg;\fg)$ is spanned by the following
cocycles:\footnotesize
\[
\begin{array}{ll}
c_{-6}=& x_1\otimes dx_{16} +y_2\otimes dx_{14} +y_6\otimes
dx_{11} +y_9\otimes dx_9 +y_{11}\otimes dx_6 +y_{14}\otimes dx_2 +y_{16}\otimes dy_1 \\

c_{-4}=& x_2\otimes dx_{14} +x_5\otimes dx_{15} +y_3\otimes dx_{10}
+y_7\otimes dx_7 +y_{10}\otimes dx_3 +y_{14}\otimes dy_2 +y_{15}\otimes dy_5 \\

c_{-2}=& x_3\otimes dx_{10} +x_6\otimes dx_{11} +x_8\otimes dx_{13}
+y_4\otimes dx_4 +y_{10}\otimes dy_3 +y_{11}\otimes dy_6
+y_{13}\otimes dy_8
\end{array}
\]
\normalsize
\end{Lemma}

\parbegin{Lemma} For $\fg=\fo\fo^{(1)}_{I\Pi}(1|8)$, $\fsl(1|4)$,
$\fsl(3|2)$, $\fsl(5)$, $\fwk(4,a)$, we have $H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p=3$}

\parbegin{Lemma} For $\fg=\fg(4,3)$, $\fg(3,6)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\parbegin{Lemma} a) For $\fg=\fg(3,3)$, with the following Cartan matrix and basis
\begin{equation}
\begin{pmatrix}
 2 & -1 & 0 & 0 \\
 -1 & 2 & -1 & 0 \\
 0 & -2 & 2 & -1 \\
 0 & 0 & 1 & 0\end{pmatrix}\ {\footnotesize\begin{array}{l}
 x_1,\ x_2,\ x_3,\ x_4,\\
x_5=[x_1, x_2],\ x_6=[x_2, x_3],\ x_7=[x_3, x_4],\\
x_8=[x_3, [x_1, x_2]],x_9=[x_3, [x_2, x_3]],x_{10}=[x_4, [x_2, x_3]],\\
x_{11}=[x_3, [x_3, [x_1, x_2]]], x_{12}=[[x_1, x_2], [x_3, x_4]],
x_{13}=[[x_2,
x_3], [x_3, x_4]], \\
x_{14}=[[x_2, x_3], [x_3, [x_1, x_2]]], x_{15}=[[x_3, x_4], [x_3,
[x_1, x_2]]], \\
x_{16}=[[x_3, [x_1, x_2]], [x_4, [x_2, x_3]]], \\
x_{17}=[[x_4, [x_2, x_3]], [x_3, [x_3, [x_1, x_2]]]]\end{array}
}\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycle:
\[
\begin{array}{l}
c_0=2\  h_1\otimes dh_5 +h_2\otimes dh_5 +h_4\otimes dh_5
\end{array}
\]

b) For $\fg=\fg(3,3)^{(1)}/\fc$, the space $H^1(\fg;\fg)$ is spanned
by the following cocycles:
\[\footnotesize
\begin{array}{ll}
c_{-8}=& y_4\otimes d x_{17} + y_7\otimes d x_{16} + 2\
y_{10}\otimes d x_{15} +y_{12}\otimes d x_{13} +y_{13}\otimes d
x_{12} + 2\  y_{15}\otimes d
x_{10}+y_{16}\otimes d x_7 +y_{17}\otimes d x_4 \\

c_0=& 2\  x_4\otimes d x_4 + 2\  x_7\otimes d x_7 + 2\ x_{10}\otimes
d x_{10} + 2\  x_{12}\otimes d x_{12} + 2\
x_{13}\otimes d x_{13} + 2\  x_{15}\otimes d x_{15} + \\

&2\  x_{16}\otimes d x_{16} + 2\  x_{17}\otimes d x_{17} +
y_4\otimes d y_4 +y_7\otimes d y_7 +y_{10}\otimes d y_{10}
+y_{12}\otimes d y_{12} +y_{13}\otimes d y_{13} +\\

&y_{15}\otimes d y_{15} +y_{16}\otimes d y_{16} +y_{17}\otimes d
y_{17}
\end{array}
\]
\end{Lemma}

\ssec{Rank=5}

\sssec{$p=2$}

\parbegin{Lemma} (a) For $\fg=\fo \fc(1;10)\subplus \Kee I_0$, $\fo
\fo \fc(1;8|2)\subplus \Kee I_0$, and $\fo\fo \fc(1;6|4)\subplus
\Kee I_0$, as well as $\fg=\mathfrak{pec}(1;5)\subplus \Kee I_0$,
with the following Cartan matrix and basis
\begin{equation}
\begin{pmatrix}
 \ev & 0 & 1 & 0 & 0 \\
 0 & \ev & 1 & 0 & 0 \\
 1 & 1 & \ev & 1 & 0 \\
 0 & 0 & 1 & \ev & 1 \\
 0 & 0 & 0 & 1 & \ev\end{pmatrix} \ {\footnotesize
\begin{array}{l}
x_1,x_2,x_3,x_4,x_5,\\
x_6=[x_1, x_3], x_7=[x_2, x_3],x_8=[x_3, x_4],x_9=[x_4,
x_5],\\
x_{10}=[x_2, [x_1, x_3]],x_{11}=[x_4, [x_1, x_3]],x_{12}=[x_4, [x_2,
x_3]],x_{13}=[x_5, [x_3, x_4]],\\
x_{14}=[x_4, [x_2, [x_1, x_3]]],x_{15}=[[x_1, x_3], [x_4,
x_5]],x_{16}=[[x_2, x_3], [x_4, x_5]],\\
x_{17}=[[x_3, x_4], [x_2, [x_1, x_3]]],x_{18}=[[x_4, x_5], [x_2,
[x_1, x_3]]],\\
x_{19}=[[x_2, [x_1, x_3]], [x_5, [x_3, x_4]]],\\
x_{20}=[[x_5, [x_3,
x_4]], [x_4,[x_2, [x_1, x_3]]]]
\end{array}
}\end{equation} the space $H^1(\fg;\fg)$ is spanned by the following
cocycle:
\[
\begin{array}{ll}
c_0=h_1\otimes dh_6 + h_2\otimes dh_6

\end{array}
\]
(b) For $\fg=\fo (10)/\fc$, $\fo \fo (8|2)/\fc$, and $\fo\fo
(6|4)/\fc$, as well as $\fg=\mathfrak{pe}(5)/\fc$, the space
$H^1(\fg;\fg)$ is spanned by the following cocycles: \footnotesize
\[
\begin{array}{ll}
c_{-8}=& y_5\otimes d x_{20} +y_9\otimes d x_{19} +y_{13}\otimes d
x_{18} +y_{15}\otimes d x_{16} +y_{16}\otimes d x_{15}
+y_{18}\otimes d x_{13}
+y_{19}\otimes d x_9 +y_{20}\otimes d x_5 \\[3mm]

c_{-6}=& x_5\otimes d x_{20} +y_4\otimes d x_{17} +y_8\otimes d
x_{14} +y_{11}\otimes d x_{12} +y_{12}\otimes d x_{11}
+y_{14}\otimes d x_8
+y_{17}\otimes d x_4 +y_{20}\otimes d y_5 \\[3mm]

c_{-4}=& x_4\otimes d x_{17} +x_9\otimes d x_{19} +y_3\otimes d
x_{10} +y_6\otimes d x_7 +y_7\otimes d x_6 +y_{10}\otimes d x_3
+y_{17}\otimes d y_4 +y_{19}\otimes d y_9 \\[3mm]

c_{-2}=& x_3\otimes d x_{10} +x_8\otimes d x_{14} +x_{13}\otimes d
x_{18} +y_1\otimes d x_2 +y_2\otimes d x_1 +y_{10}\otimes d y_3
+y_{14}\otimes d y_8 +y_{18}\otimes d y_{13} \\[3mm]

c_{0}^1=& x_2\otimes d x_1 +x_7\otimes d x_6 +x_{12}\otimes d x_{11}
+x_{16}\otimes d x_{15} +y_1\otimes d y_2 +y_6\otimes d y_7
+y_{11}\otimes d y_{12} +y_{15}\otimes d y_{16} \\

c_{0}^2=& x_1\otimes d x_2 +x_6\otimes d x_7 +x_{11}\otimes d x_{12}
+x_{15}\otimes d x_{16}
+y_2\otimes d y_1 +y_7\otimes d y_6 +y_{12}\otimes d y_{11} +y_{16}\otimes d y_{15} \\

c_{0}^3=& x_2\otimes d x_2 +x_4\otimes d x_4 +x_7\otimes d x_7
+x_8\otimes d x_8 +x_9\otimes d x_9 +x_{10}\otimes d x_{10}
+x_{11}\otimes d x_{11} +x_{13}\otimes d x_{13} +\\

&x_{15}\otimes d x_{15} +x_{20}\otimes d x_{20} +y_2\otimes d y_2
+y_4\otimes d y_4 +y_7\otimes d y_7 +y_8\otimes d y_8 +y_9\otimes d
y_9 +y_{10}\otimes d y_{10} +\\

&y_{11}\otimes d y_{11} +y_{13}\otimes d y_{13} +y_{15}\otimes d
y_{15} +y_{20}\otimes d y_{20}
 \end{array}
 \]
\normalsize
\end{Lemma}



%\parbegin{Lemma} (a) For $\fg=\fo_\Pi^1(11)$, with the following Cartan matrix and basis
%\begin{equation}
%\begin{pmatrix}
% \od & 1 & 0 & 0 & 0 \\
% 1 & \ev & 1 & 0 & 0 \\
% 0 & 1 & \ev & 1 & 0 \\
% 0 & 0 & 1 & \ev & 1 \\
% 0 & 0 & 0 & 1 & \ev\end{pmatrix} \ {\footnotesize
%\begin{array}{l}
%x_1,x_2,x_3,x_4,x_5, \\
%x_6=[x_1, x_2],x_7=[x_2, x_3],x_8=[x_3, x_4],x_9=[x_4,
%x_5],\\
%x_{10}=[x_1, [x_1, x_2]],x_{11}=[x_3, [x_1, x_2]],\\
%\hskip 1cm x_{12}=[x_4, [x_2,
%x_3]],x_{13}=[x_5, [x_3, x_4]],\\
%x_{14}=[x_3, [x_1, [x_1, x_2]]],x_{15}=[[x_1, x_2], [x_3,
%x_4]],x_{16}=[[x_2, x_3], [x_4, x_5]],\\
%x_{17}=[[x_2, x_3], [x_1, [x_1, x_2]]],x_{18}=[[x_3, x_4], [x_1,
%[x_1, x_2]]],\\
%\hskip 1cm x_{19}=[[x_4, x_5], [x_3, [x_1, x_2]]],\\
%x_{20}=[[x_1, [x_1, x_2]], [x_4, [x_2, x_3]]],x_{21}=[[x_1, [x_1,
%x_2]], [x_5, [x_3, x_4]]],\\
%x_{22}=[[x_1, [x_1, x_2]], [[x_2, x_3], [x_4, x_5]]],\\
%\hskip 1cm x_{23}=[[x_4,
%[x_2, x_3]], [x_3, [x_1, [x_1, x_2]]]],\\
%x_{24}=[[x_3, [x_1, [x_1, x_2]]], [[x_2, x_3], [x_4,
%x_5]]],\\
%x_{25}=[[[x_2, x_3], [x_4, x_5]], [[x_3, x_4], [x_1, [x_1,
%x_2]]]]\end{array} }
%\end{equation} the space $H^1(\fg;\fg)$ is
%spanned by the following cocycles: \footnotesize
%\[
%\begin{array}{ll}
%c_{-10}=& y_5\otimes dx_{25} + y_9\otimes dx_{24} + y_{13}\otimes
%dx_{22} + y_{16}\otimes dx_{21} + y_{19}\otimes dx_{19} +
%y_{21}\otimes dx_{16}
%+ y_{22}\otimes dx_{13}\\

%& + y_{24}\otimes dx_9 + y_{25}\otimes dx_5 \\

%c_{-8}=& x_5\otimes dx_{25} +y_4\otimes dx_{23} + y_8\otimes dx_{20}
%+y_{12}\otimes dx_{18} + y_{15}\otimes dx_{15}
%+y_{18}\otimes dx_{12} +y_{20}\otimes dx_8 \\

%& + y_{23}\otimes dx_4
%+y_{25}\otimes dy_5 \\

%c_{-6}=& x_4\otimes dx_{23} +x_9\otimes dx_{24} + y_3\otimes dx_{17}
%+y_7\otimes dx_{14} + y_{11}\otimes dx_{11} +y_{14}\otimes d
%x_7 +y_{17}\otimes dx_3 \\

%& +
%y_{23}\otimes dy_4 +y_{24}\otimes dy_9 \\

%c_{-4}=& x_3\otimes dx_{17} +x_8\otimes dx_{20} + x_{13}\otimes
%dx_{22} +y_2\otimes dx_{10} + y_6\otimes dx_6 +y_{10}\otimes dx_2
%+y_{17}\otimes dy_3 \\

%& +y_{20}\otimes dy_8 +
%y_{22}\otimes dy_{13} \\

%c_{-2}=& x_2\otimes dx_{10} +x_7\otimes dx_{14} + x_{12}\otimes
%dx_{18} +x_{16}\otimes dx_{21} + y_1\otimes dx_1 +y_{10}\otimes d
%y_2 +y_{14}\otimes dy_7 \\

%& +y_{18}\otimes dy_{12} +y_{21}\otimes dy_{16}
% \end{array}
% \]
 %\normalsize
%\end{Lemma}

\parbegin{Lemma} (a) For $\fg=\fgl(6)$, as well as $\fg=\fgl(5|1)$,
$\fgl(4|2)$, $\fgl(3|3)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycle (here $h_6$ is the outer derivation; same
situation as for $\fgl(3)$ for $p=3$):
\[
c_{0}=h_1\otimes dh_6 + h_3\otimes dh_6 + h_5\otimes dh_6
\]

(b) For $\fg=\fpsl(6)$, as well as $\fpsl(5|1)$, $\fpsl(4|2)$,
$\fpsl(3|3)$, the space $H^1(\fg;\fg)$ is spanned by the following
cocycle turning $\fpsl$ into $\fpgl$:
\[
\begin{array}{ll}
c_{0}=&x_3\otimes d x_3+ x_4\otimes d x_4+ x_7\otimes d x_7+
x_9\otimes d x_9+ x_{10}\otimes d x_{10}+ y_3\otimes d y_3+\\

&y_4\otimes d y_4+ y_7\otimes d y_7+ y_9\otimes d y_9+ y_{10}\otimes
d y_{10}
\end{array}
\]
\end{Lemma}




\parbegin{Lemma} (a) For $\fg=\fo\fo^{(1)}_{I\Pi}(9|2)$,
$\fo\fo^{(1)}_{I\Pi}(7|4)$, $\fo\fo^{(1)}_{I\Pi}(5|6)$, and
$\fo\fo^{(1)}_{I\Pi}(3|8)$ with the following Cartan matrix and
basis
\begin{equation}
\begin{pmatrix}
 \ev & 1 & 0 & 0 & 0 \\
 1 & 0 & 1 & 0 & 0 \\
 0 & 1 & \ev & 1 & 0 \\
 0 & 0 & 1 & \ev & 1 \\
 0 & 0 & 0 & 1 &\od \end{pmatrix} \quad
{\tiny
\begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\ x_5,\\
x_6=[x_1, x_2],\ x_7=[x_2, x_3],\ x_8=[x_3, x_4],\ x_9=[x_4,
x_5],\\
x_{10}=[x_3, [x_1, x_2]],\ x_{11}=[x_4, [x_2, x_3]],\ x_{12}=[x_5,
[x_3,
x_4]],\\
\hskip 0.5cm x_{13}=[x_5, [x_4, x_5]],\\
x_{14}=[x_5, [x_5, [x_3, x_4]]],\ x_{15}=[[x_1, x_2], [x_3,
x_4]],\ x_{16}=[[x_2, x_3], [x_4, x_5]],\\
x_{17}=[[x_2, x_3], [x_5, [x_4, x_5]]],\ x_{18}=[[x_4, x_5], [x_3,
[x_1, x_2]]],\\
\hskip 0.5cm x_{19}=[[x_4, x_5], [x_5, [x_3, x_4]]],\\
x_{20}=[[x_3, [x_1, x_2]], [x_5, [x_4, x_5]]],x_{21}=[[x_4, [x_2,
x_3]], [x_5, [x_4, x_5]]],\\
x_{22}=[[x_5, [x_3, x_4]], [[x_2, x_3], [x_4, x_5]]],\\
\hskip 0.5cm x_{23}=[[x_5,
[x_4, x_5]], [[x_1, x_2], [x_3, x_4]]],\\
x_{24}=[[x_5, [x_5, [x_3, x_4]]], [[x_1, x_2], [x_3, x_4]]],\
x_{25}=[[x_2, x_3], [x_4,
x_5]]{}^2,\\
x_{26}=[[[x_2, x_3], [x_4, x_5]], [[x_4, x_5], [x_3, [x_1,
x_2]]]],\\
x_{27}=[[x_4, x_5], [x_3, [x_1, x_2]]]{}^2\end{array} }
\end{equation}
the space $H^1(\fg;\fg)$ is spanned by the following cocycles:
\[
{\tiny\begin{array}{ll} c_{-8}=& x_1\otimes dx_{25} +y_2\otimes
dx_{22} +y_7\otimes dx_{21} +y_{11}\otimes dx_{17} +y_{16}\otimes
dx_{16} + y_{17}\otimes
dx_{11}\\

& +y_{21}\otimes dx_7 +
y_{22}\otimes dx_2 +y_{25}\otimes dy_1 \\
c_{-6}=& x_2\otimes dx_{22} +x_6\otimes dx_{24} + y_3\otimes dx_{19}
+y_8\otimes dx_{14} + y_{12}\otimes dx_{12} +y_{14}\otimes d
x_8\\
& +y_{19}\otimes dx_3 +
y_{22}\otimes dy_2 +y_{24}\otimes dy_6 \\

c_{-4}=& x_3\otimes dx_{19} +x_7\otimes dx_{21} + x_{10}\otimes
dx_{23} +y_4\otimes dx_{13} + y_9\otimes dx_9 +y_{13}\otimes dx_4\\
& + y_{19}\otimes dy_3 +y_{21}\otimes dy_7 +
y_{23}\otimes dy_{10} \\

c_{-2}=& x_4\otimes dx_{13} +x_8\otimes dx_{14} + x_{11}\otimes
dx_{17} +x_{15}\otimes dx_{20} + y_5\otimes dx_5 +y_{13}\otimes d
y_4\\
& +y_{14}\otimes dy_8 + y_{17}\otimes dy_{11} +y_{20}\otimes dy_{15}
\end{array}}
\]
\end{Lemma}

\parbegin{Lemma} For $\fg=\fo\fo^{(1)}_{I\Pi}(1|10)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}


\sssec{$p=3$}

\parbegin{Lemma} (a) For $\fg=\fgl(6)$, $\fgl(4|2)$, and
$\fgl(3|3)$, the space $H^1(\fg;\fg)$ is spanned by the following
cocycle (here $h_6$ is the outer derivation; same situation as for
$\fgl(3)$ for $p=3$):
\[
c_{0}=2\  h_1\otimes dh_6 + h_2\otimes dh_6 + 2\  h_4\otimes dh_6 +
h_5\otimes dh_6
\]

(b) For $\fg=\fpsl(6)$, the space $H^1(\fg;\fg)$ is spanned by the following cocycle
turning $\fpsl$ into $\fpgl$:
\[
\begin{array}{ll}
c_{0}=&2x_3\otimes d x_3+ x_4\otimes d x_4+ 2x_7\otimes d x_7+
x_9\otimes d x_9+ 2x_{10}\otimes d x_{10}+ y_3\otimes d y_3\\
&+ 2y_4\otimes d y_4+ y_7\otimes d y_7+ 2y_9\otimes d y_9+
y_{10}\otimes d y_{10}
\end{array}
\]

(c) For $\fg=\fpsl(3|3)$, the space $H^1(\fg;\fg)$ is spanned by the following cocycle
turning $\fpsl$ into $\fpgl$:
\[
\begin{array}{ll}
c_{0}=&2x_3\otimes d x_3+ 2x_4\otimes d x_4+ 2x_7\otimes d x_7+
2x_9\otimes d x_9+ 2x_{10}\otimes d x_{10}+ y_3\otimes d y_3\\
&+ y_4\otimes d y_4+ y_7\otimes d y_7+ y_9\otimes d y_9+
y_{10}\otimes d y_{10}
\end{array}
\]
 \end{Lemma}


\parbegin{Lemma} a) For $\fg=\fg(8,3), \fg=\mathfrak{el}(5;3)$, we have
$H^1(\fg;\fg)=0$.

b) For $\fg=\fg(2,6)$ with Cartan matrix and basis
\[
\begin{pmatrix}
 2 & -1 & 0 & 0 & 0 \\
 -1 & 2 & -1 & 0 & 0 \\
 0 & -1 & 2 & -1 & -1 \\
 0 & 0 & -1 & 2 & 0 \\
 0 & 0 & 1 & 0 & 0
\end{pmatrix}
\ \ \footnotesize{\begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\ x_5,\\
x_6=[x_1, x_2],\ x_7=[x_2, x_3],\ x_8=[x_3, x_4],\ x_9=[x_3,
x_5],\\
x_{10}=[x_3, [x_1, x_2]],\ x_{11}=[x_4, [x_2, x_3]],\ x_{12}=[x_5,
[x_2, x_3]],\ x_{13}=[x_5, [x_3, x_4]],\\
x_{14}=[x_5, [x_4, [x_2, x_3]]],\ x_{15}=[[x_1, x_2], [x_3, x_4]],\
x_{16}=[[x_1, x_2], [x_3, x_5]], \\
 x_{17}=[[x_1, x_2], [x_5, [x_3,
x_4]]], \ x_{18}=[[x_3, x_5], [x_4, [x_2, x_3]]],\\
x_{19}=[[x_3, [x_1, x_2]], [x_5, [x_3, x_4]]], \ x_{20}=[[x_5, [x_2,
x_3]], [x_5, [x_3, x_4]]], \\
x_{21}= [[x_5, [x_2, x_3]], [[x_1, x_2], [x_3, x_4]]],\
[x_{22}=[[x_5, [x_3, x_4]], [[x_1, x_2], [x_3, x_5]]], \\
x_{23}= [[x_5, [x_4, [x_2, x_3]]], [[x_1, x_2], [x_3, x_5]]],\\
x_{24}=[[[x_1, x_2], [x_3, x_5]], [[x_3, x_5], [x_4, [x_2,
x_3]]]],\\
x_{25}=[[[x_1, x_2], [x_5, [x_3, x_4]]], [[x_3, x_5], [x_4, [x_2,
x_3]]]]
\end{array}
}
\]
and grading operator $h_6=(1,0,0,0,0)$, the space $H^1(\fg;\fg)$ is
spanned by the cocycle
\[
c_0=2\  h_1\otimes dh_6+ h_2\otimes dh_6+ h_5\otimes dh_6
\]
For $\fg(2,6)^{(1)}/\fc$, the space $H^1(\fg;\fg)$ is spanned by the
cocycle
\[
\footnotesize{\begin{array}{ll} c_0=& 2\  x_4\otimes d x_4 +2\
x_5\otimes d x_5 +2\  x_8\otimes d x_8 +2\  x_9\otimes d x_9 +2\
x_{11}\otimes d x_{11} +2\  x_{12}\otimes d x_{12} +2\
x_{15}\otimes d x_{15} +\\

&2\  x_{16}\otimes d x_{16} +x_{20}\otimes d x_{20} +x_{22}\otimes d
x_{22} +x_{23}\otimes d x_{23} +x_{24}\otimes d x_{24} +y_4\otimes d
y_4 +y_5\otimes d y_5 +y_8\otimes d y_8 +\\

&y_9\otimes d y_9 +y_{11}\otimes d y_{11} +y_{12}\otimes d y_{12}
+y_{15}\otimes d y_{15} +y_{16}\otimes d y_{16} +2\  y_{20}\otimes d
y_{20} +\\

&2\  y_{22}\otimes d y_{22} +2\  y_{23}\otimes d y_{23} +2\
y_{24}\otimes d y_{24}
\end{array}
}
\]
\end{Lemma}




\sssec{$p=5$}

\parbegin{Lemma} (a) For $\fg=\fgl(3|3)$, the space $H^1(\fg;\fg)$
is spanned by the following cocycle (here $h_6$ is the outer
derivation; same situation as for $\fgl(3)$ for $p=3$):
\[
c_{0}=4\  h_1\otimes dh_6 + 3\  h_2\otimes dh_6 + 2\  h_3\otimes
dh_6 + 2\  h_4\otimes dh_6 + h_5\otimes dh_6
\]

(b) For $\fg=\fpsl(3|3)$, the space $H^1(\fg;\fg)$ is spanned by the
following cocycle turning $\fpsl$ into $\fpgl$:
\[
\begin{array}{ll}
c_{0}=&4\  x_2\otimes d x_2+ 4\  x_3\otimes d x_3+ 4\  x_6\otimes d
x_6+ 4\  x_8\otimes d x_8+ 4\  x_{12}\otimes d x_{12}+ y_2\otimes d
y_2\\
&+ y_3\otimes d y_3+ y_6\otimes d y_6+ y_8\otimes d y_8+
y_{12}\otimes d y_{12}
\end{array}
\]
 \end{Lemma}

\parbegin{Lemma} For $\fg=\mathfrak{el}(5;5)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\ssec{Rank=6}

\sssec{$p=2$}

\parbegin{Lemma} For $\fg=\fe (6;1)$, $\fe (6,6)$, we
have $H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p=3$}

\parbegin{Lemma} For $\fg=\mathfrak{g}(4;8)$, $\mathfrak{g}(6;6)$, we
have $H^1(\fg;\fg)=0$.
\end{Lemma}

\ssec{Rank=7}

\sssec{$p=2$}


\parbegin{Lemma} Let the Chevalley basis of $\fg=\fe(7)$ (those of $\fg=\fe(7,i)$ differ
in that $p(x_i)=p(y_i)=\od$, the other Chevalley generators being
even) be
\[
{\tiny\begin{array}{l} x_1,\
x_2,\  x_3,\ x_4,\ x_5,\ x_6,\ x_7,\\
x_8=[x_1, x_2], \ x_9= [x_2, x_3],\ x_{10}=[x_3, x_4], \
x_{11}=[x_4,
x_5], \ x_{12}=[x_4, x_7],\ x_{13}=[x_5, x_6], \\

x_{14}=[x_3, [x_1, x_2]], \ x_{15}=[x_4, [x_2, x_3]], \ x_{16}=[x_5,
[x_3, x_4]],\ x_{17}=[x_6, [x_4, x_5]],\\

\hskip .5cm x_{18}=[x_7, [x_3, x_4]],\
x_{19}=[x_7, [x_4, x_5]], \\

x_{20}=[x_7, [x_5, [x_3, x_4]]], \ x_{21}=[[x_1, x_2], [x_3, x_4]],\
x_{22}=[[x_2, x_3], [x_4, x_5]], \ \\

\hskip .5cm x_{23}=[[x_2, x_3], [x_4, x_7]],\ x_{24}= [[x_3, x_4],
[x_5, x_6]],\ x_{25}=[[x_4, x_7], [x_5,
x_6]],\\
x_{26}=[[x_2, x_3], [x_7, [x_4, x_5]]], \ x_{27}=[[x_4, x_5], [x_3,
[x_1, x_2]]],\ x_{28}=[[x_4, x_7], [x_3, [x_1, x_2]]],\ \\

\hskip .5cm x_{29}= [[x_4, x_7], [x_5, [x_3, x_4]]],\ x_{30}=[[x_5,
x_6], [x_4, [x_2,
x_3]]],\ x_{31}=[[x_5, x_6], [x_7, [x_3, x_4]]], \\

x_{32}= [[x_5, x_6], [[x_2, x_3], [x_4, x_7]]],\ x_{33}=[[x_3, [x_1,
x_2]], [x_6, [x_4, x_5]]],\ x_{34}=[[x_3, [x_1, x_2]] , [x_7, [x_4,
x_5]]],\ \\

\hskip .5cm x_{35}=[[x_4, [x_2, x_3]], [x_7, [x_4, x_5]]],\
x_{36}=[[x_6, [x_4, x_5]], [x_7, [x_3, x_4]]], \\

x_{37}=[[x_3, [x_1, x_2]] , [[x_4, x_7], [x_5, x_6]]],\
x_{38}=[[x_6, [x_4, x_5]], [[x_2, x_3] , [x_4, x_7]]],\
x_{39}=[[x_7, [x_3, x_4]], [[x_2, x_3], [x_4, x_5]]],\ \\

\hskip .5cm x_{40}=[[x_7, [x_4, x_5]], [[x_1, x_2], [x_3, x_4]]], \
x_{41}= [[x_7, [x_4, x_5]]
, [[x_3, x_4], [x_5, x_6]]],\\

 x_{42}= [[x_7, [x_5, [x_3, x_4]]],
[[x_1, x_2], [x_3, x_4]]], \ x_{43}= [[[x_1, x_2], [x_3, x_4]],
[[x_4, x_7], [x_5, x_6]]],\ \\

\hskip .5cm x_{44}=[[[x_2, x_3], [x_4, x_5]], [[x_4, x_7], [x_5,
x_6]]],\ x_{45}=[[[x_2, x_3], [x_4, x_7]], [[x_3, x_4],
[x_5, x_6]]],\\

 x_{46}=[[[x_2, x_3], [x_4, x_7]], [[x_4, x_5], [x_3,
[x_1, x_2]]]],\ x_{47}=[[[x_3, x_4], [x_5, x_6]] , [[x_2, x_3],
[x_7, [x_4, x_5]]]],\ \\

\hskip .5cm x_{48}=[[[x_3, x_4], [x_5, x_6]], [[x_4, x_7], [x_3,
[x_1, x_2]]]], \ x_{49}=[[[x_4, x_7] , [x_5, x_6]], [[x_4,
x_5], [x_3, [x_1, x_2]]]],\\

 x_{50}=[[[x_4, x_5], [x_3, [x_1,
x_2]]], [[x_5, x_6], [x_7, [x_3, x_4]]]], \ x_{51}=[[[x_4, x_7],
[x_3, [x_1, x_2]]], [[x_5, x_6], [x_4, [x_2, x_3]]]],\\

x_{52}= [[[x_4, x_7], [x_5, [x_3, x_4]]], [[x_5, x_6], [x_4, [x_2,
x_3]]]], \\


 x_{53}=[[[x_4, x_7] , [x_5, [x_3, x_4]]], [[x_3, [x_1,
x_2]], [x_6, [x_4, x_5]]]],\ x_{54}= [[[x_5, x_6], [x_4, [x_2,
x_3]]], [[x_3, [x_1, x_2]], [x_7, [x_4, x_5]]]], \ \\

\hskip .5cm x_{55}=[[[x_5, x_6], [x_7, [x_3, x_4]]], [[x_4, [x_2,
x_3]], [x_7, [x_4, x_5]]]],
\\

x_{56}=[[[x_3, [x_1, x_2]], [x_6, [x_4, x_5]]], [[x_4, [x_2, x_3]],
[x_7, [x_4, x_5]]]], \ \\

\hskip .5cm x_{57}=[[[x_3, [x_1, x_2]] , [x_7, [x_4,
x_5]]], [[x_6, [x_4, x_5]], [x_7, [x_3, x_4]]]],\\

 x_{58}=[[[x_3,
[x_1, x_2]], [x_6, [x_4, x_5]]], [[x_7, [x_3, x_4]], [[x_2, x_3],
[x_4, x_5]]]], \\

\hskip .5cm x_{59}= [[[x_4, [x_2, x_3]], [x_7, [x_4, x_5]]],
[[x_3, [x_1, x_2]], [[x_4, x_7], [x_5, x_6]]]],\\


 x_{60}=[[[x_3, [x_1, x_2]]
, [[x_4, x_7], [x_5, x_6]]], [[x_7, [x_3, x_4]] , [[x_2, x_3], [x_4,
x_5]]]],\\

 x_{61}= [[[x_7, [x_4, x_5]], [[x_1, x_2], [x_3, x_4]]],
[[[x_2, x_3], [x_4, x_7]] , [[x_3, x_4], [x_5, x_6]]]],\\

 x_{62}=
[[[x_7, [x_5, [x_3, x_4]]] , [[x_1, x_2], [x_3, x_4]]], [[[x_2,
x_3], [x_4, x_5]], [[x_4, x_7], [x_5, x_6]]]],\\


 x_{63}= [[[[x_2,
x_3], [x_4, x_7]], [[x_3, x_4], [x_5, x_6]]], [[[x_4, x_7] , [x_5,
x_6]], [[x_4, x_5], [x_3, [x_1, x_2]]]]]\end{array} }
\]
(a) For $\fg=\fe(7,i)$, where $i=1, 6$ or $7$, the space
$H^1(\fg;\fg)$ is spanned by the following cocycle:
\[
c_{0}=h_1\otimes dh_8 + h_3\otimes dh_8 + h_7\otimes dh_8
\]

(b) For $\fg=\fe^{(1)}(7,i)/\fc $, where $i=1, 6$ or $7$, the space
$H^1(\fg;\fg)$ is spanned by the following cocycle
\[
\footnotesize{\begin{array}{ll} c_0=& x_2\otimes d x_2 +x_4\otimes d
x_4 +x_7\otimes d x_7 +x_8\otimes d x_8 +x_9\otimes
d x_9 +x_{10}\otimes d x_{10} +x_{11}\otimes d x_{11} +x_{14}\otimes d x_{14} \\

& +x_{16}\otimes d x_{16} +x_{17}\otimes d x_{17} +x_{23}\otimes d
x_{23} +x_{24}\otimes d x_{24}
+x_{26}\otimes d x_{26} +x_{28}\otimes d x_{28} +x_{29}\otimes d x_{29}\\

& +x_{32}\otimes d x_{32} +x_{34}\otimes d x_{34} +x_{36}\otimes d
x_{36} +x_{37}\otimes d x_{37}
+x_{41}\otimes d x_{41} +x_{46}\otimes d x_{46} +x_{51}\otimes d x_{51} \\

& +x_{52}\otimes d x_{52} +x_{53}\otimes d x_{53} +x_{54}\otimes d
x_{54} +x_{59}\otimes d x_{59}
+x_{60}\otimes d x_{60} +y_2\otimes d y_2 +y_4\otimes d y_4 \\

& +y_7\otimes d y_7 +y_8\otimes d y_8 +y_9\otimes d y_9
+y_{10}\otimes d y_{10} +y_{11}\otimes d y_{11} +y_{14}\otimes d y_{14} +y_{16}\otimes d y_{16} \\

& +y_{17}\otimes d y_{17} +y_{23}\otimes d y_{23} +y_{24}\otimes d
y_{24} +y_{26}\otimes d y_{26}
+y_{28}\otimes d y_{28} +y_{29}\otimes d y_{29} +y_{32}\otimes d y_{32} \\

& +y_{34}\otimes d y_{34} +y_{36}\otimes d y_{36} +y_{37}\otimes d
y_{37} +y_{41}\otimes d y_{41}
+y_{46}\otimes d y_{46} +y_{51}\otimes d y_{51} +y_{52}\otimes d y_{52} \\

& +y_{53}\otimes d y_{53} +y_{54}\otimes d y_{54} +y_{59}\otimes d
y_{59} +y_{60}\otimes d y_{60}
\end{array}
}\]

\end{Lemma}


\sssec{$p=3$}

\parbegin{Lemma}  For $\fg=\mathfrak{g}(8;8)$, we have
$H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{Rank=8}

\sssec{$p=2$}


\parbegin{Lemma} For $\fg=\fe (8), \fe (8;1)$, $\fe(8,8)$, we have $H^1(\fg;\fg)=0$.
\end{Lemma}

\sssec{$p>2$}


\parbegin{Lemma} For $\fg=\fe (8)$, we have $H^1(\fg;\fg)=0$.
\end{Lemma}



\ssec{Symmetric queerifications}

\sssbegin{Lemma} For $\fg=\mathfrak{q}^{(1)}(\fwk(3;a)/\fc)$, see
\eqref{wk3a}, the space $H^1(\fg;\fg)$ is spanned by the following
three cocycles: \footnotesize
\[
\begin{array}{ll}
c_{0}^1=&h_1\otimes d (\Pi(h_1))+ h_2\otimes d (\Pi(h_2))+
x_1\otimes d (\Pi(x_1)) + x_5\otimes d (\Pi(x_5))+ x_6\otimes d
(\Pi(x_6)) +
y_1\otimes d (\Pi(y_1))+ \\

&y_5\otimes d (\Pi(y_5))+ y_6\otimes d (\Pi(y_6))+ \Pi(x_2)\otimes d
x_2+ \Pi(x_3) \otimes d x_3 + \Pi(x_4)\otimes d (x_4)+
\Pi(x_7)\otimes d x_7+\\

&\Pi(y_2)\otimes
d y_2+ \Pi(y_3)\otimes d y_3+ \Pi(y_4)\otimes d y_4+ \Pi(y_7)\otimes d (y_7)\\[3mm]

c_{0}^2=&x_3\otimes d x_3+ x_5\otimes x_5+ x_6\otimes d x_6+
x_7\otimes d x_7+ y_3\otimes d y_3+ y_5\otimes d y_5+ y_6\otimes d
y_6+
y_7\otimes d (y_7)+ \\

&\Pi(h_1)\otimes d(\Pi(h_1))+ \Pi(h_2)\otimes d (\Pi(h_2))+
\Pi(x_1)\otimes d (\Pi(x_1))+ \Pi(x_2)\otimes d (\Pi(x_2))+
\Pi(x_4)\otimes d
(\Pi(x_4))+ \\

&\Pi(y_1)\otimes d (\Pi(y_1))+ \Pi(y_2)\otimes d(\Pi(y_2))
+ \Pi(y_4)\otimes d(\Pi(y_4))\\[3mm]

c_{0}^3=&h_1\otimes d(\Pi(h_1))+ h_2\otimes d (\Pi(h_2))+ x_1\otimes
d (\Pi(x_1)) + x_2\otimes d (\Pi(x_2))+ x_3\otimes d (\Pi(x_3)) +
x_4\otimes d (\Pi(x_4))+ \\

&x_5\otimes d (\Pi(x_5))+ x_6\otimes d (\Pi(x_6))+ x_7\otimes d
(\Pi(x_7))+ y_1\otimes d (\Pi(y_1)) + y_2\otimes d(\Pi(y_2))+
y_3\otimes d (\Pi(y_3))+ \\

&y_4\otimes d(\Pi(y_4))+ y_5\otimes d (\Pi(y_5))+ y_6\otimes d
(\Pi(y_6)) + y_7\otimes d (\Pi(y_7))
\end{array}
\]
\normalsize
\end{Lemma}

\sssbegin{Lemma} For $\fg=\mathfrak{q}(\fwk(4;a))$ for the following
Cartan matrix and basis of $\fwk(4;a)$
\footnotesize{\begin{equation}
\begin{pmatrix}
 0 & \alpha & 1 & 0 \\
 \alpha & 0 & 0 & 0 \\
 1 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0 \end{pmatrix} \quad
\begin{array}{l}
x_1,\ x_2,\ x_3,\ x_4,\\
x_5=[x_1, x_2], \ x_6=[x_1, x_3],\ x_7=[x_3, x_4],\\
x_8=[x_3, [x_1, x_2]],\ x_9=[x_4, [x_1, x_3]], \\
x_{10}=[[x_1, x_2], [x_1, x_3]],\ x_{11}=[[x_1, x_2], [x_3,
x_4]],\\
x_{12}=[[x_1, x_2], [x_4, [x_1, x_3]]], \\
x_{13}=[[x_3, [x_1, x_2]], [x_4, [x_1, x_3]]], \\
x_{14}=[[x_4, [x_1, x_3]], [[x_1, x_2], [x_1, x_3]]],
\\
x_{15}=[[[x_1, x_2], [x_1, x_3]], [[x_1, x_2], [x_3,
x_4]]]\end{array}
\end{equation}
} the space $H^1(\fg;\fg)$ is spanned by the following two cocycles:
\footnotesize
\[
\begin{array}{ll}
c_{0}^1=&h_1\otimes d (\Pi(y_1))+ h_2\otimes d (\Pi(h_2))+
h_3\otimes d (\Pi(h_3)+ h_4\otimes d (\Pi(h_4))+ x_1 \otimes d
(\Pi(x_1)) +
x_2\otimes d (\Pi(x_2))+ \\

&x_3\otimes d (\Pi(x_3))+ x_4\otimes d (\Pi(x_4)) + x_5\otimes d
(\Pi(x_5))+ x_6\otimes d (\Pi(x_6))
+ x_7\otimes d (\Pi(x_7)) + x_8\otimes d (\Pi(x_8))+ \\

&x_9\otimes d (\Pi(x_9))+ x_{10}\otimes d (\Pi(x_{10})) +
x_{11}\otimes d (\Pi(x_{11})) + x_{12}\otimes d (\Pi(x_{12}))+
x_{13}\otimes d(\Pi(x_{13})) + \\

&x_{14}\otimes d (\Pi(x_{14}))+ x_{15}\otimes d (\Pi(x_{15})+
y_{1}\otimes d(\Pi(y_{1})) + y_{2}\otimes d (\Pi(y_2))+ y_{3}\otimes
d (\Pi(y_3))+ y_{4}\otimes d (\Pi(y_4))+ \\

&y_{5}\otimes d (\Pi(y_5))+ y_{6}\otimes d (\Pi(y_6))+ y_{7}\otimes
d (\Pi(y_7))+ y_{8}\otimes d (\Pi(y_8))+ y_{9}\otimes d (\Pi(y_9))+
y_{10}\otimes d (\Pi(y_{10}))+ \\

&y_{11}\otimes d (\Pi(y_{11}))+y_{12}\otimes d (\Pi(y_{12}))+
y_{13}\otimes d (\Pi(y_{13})) + y_{14}\otimes d (\Pi(y_{14}))+
y_{15}\otimes d
(\Pi(y_{15}))\\[3mm]

c_{0}^2=&x_4\otimes d x_4+ x_7\otimes d x_7 + x_9\otimes d x_9+
x_{11}\otimes d x_{11}+ x_{12}\otimes d x_{12} + x_{13}\otimes d
x_{13}+
x_{14}\otimes d x_{14} + x_{15}\otimes d x_{15} + \\

&y_4\otimes d y_4+ y_7\otimes d y_7+ y_9\otimes d y_9 +
y_{11}\otimes d y_{11}+ y_{12}\otimes d y_{12}+ y_{13}\otimes d
y_{13} +
y_{14}\otimes d y_{14}+ y_{15}\otimes d y_{15}+ \\

&\Pi(h_1)\otimes d (\Pi(h_1)) + \Pi(h_2)\otimes d (\Pi(h_2))+
\Pi(h_3)\otimes d (\Pi(h_3) + \Pi(h_4)\otimes d (\Pi(h_4)) +
\Pi(x_1)\otimes d (\Pi(x_1))+ \\

&\Pi(x_2)\otimes d (\Pi(x_2))+ \Pi(x_3)\otimes d (\Pi(x_3)) +
\Pi(x_5)\otimes d (\Pi(x_5))+ \Pi(x_6)\otimes d (\Pi(x_6))+
\Pi(x_8)\otimes d (\Pi(x_8))
+ \\

&\Pi(x_{10})\otimes d (\Pi(x_{10})) + \Pi(y_1)\otimes d (\Pi(y_{1}))
+ \Pi(y_2)\otimes d (\Pi(y_2)) + \Pi(y_3)\otimes d (\Pi(y_3))+
\Pi(y_5)\otimes d (\Pi(y_5)) + \\

&\Pi(y_6)\otimes d (\Pi(y_6)) + \Pi(y_8)\otimes d (\Pi(y_8))+
\Pi(y_{10})\otimes d (\Pi(y_{10}))
 \end{array}
\]
\normalsize
\end{Lemma}

\ssec{Orthogonal algebras of the type $\fo_I$ and their derived} Let
$X_{i,j}:=E^{i,j}+E^{j,i}$.


\sssbegin{Lemma} For $\fg=\fo^{(1)}_I(n)$, where $n\neq 4$. the
space $H^1(\fg;\fg)$ is spanned by $n-1$ cocycles of degree $0$ and
the following form: \footnotesize{\[
\begin{array}{l}
X_{1,2}\otimes dX_{1,2}
+ X_{1,3}\otimes dX_{1,3} +\dots + X_{1,n}\otimes dX_{1,n}, \\
X_{1,2}\otimes dX_{1,2} + X_{2,3}\otimes dX_{2,3}+\dots +
X_{2,n}\otimes dX_{2,n},\\
X_{1,3}\otimes dX_{1,3} + X_{2,3}\otimes dX_{2,3}+\dots +
X_{3,n}\otimes dX_{3,n},\\
\dotfill\\
X_{1,n-1}\otimes dX_{1,n-1} + X_{2,n-1}\otimes dX_{2,n-1}+\dots +
X_{n-1,n}\otimes dX_{n-1,n}.
\end{array}
\]
}
\end{Lemma}

\parbegin{Lemma} For $\fg=\fo^{(1)}_I(4)$, the space $H^1(\fg;\fg)$
is spanned by the following cocycles of degree $0$: \footnotesize{\[
\begin{array}{ll}\text{regular cases}&\begin{cases}
X_{1,2}\otimes dX_{1,2} + X_{1,3}\otimes dX_{1,3}
+ X_{1,4}\otimes dX_{1,4} \\
X_{1,2}\otimes dX_{1,2} + X_{2,3}\otimes dX_{2,3}
+ X_{2,4}\otimes dX_{2,4} \\
X_{1,3}\otimes dX_{1,3} + X_{2,3}\otimes dX_{2,3} +
X_{3,4}\otimes dX_{3,4}\\[2mm]
\end{cases}\\
\text{exceptional cases}&\begin{cases}X_{1,2}\otimes dX_{3,4} +
X_{1,3}\otimes dX_{2,4}+ X_{2,3}\otimes dX_{1,4} \\
X_{1,2}\otimes dX_{3,4} + X_{1,4}\otimes dX_{2,3}
+ X_{2,4}\otimes dX_{1,3} \\
X_{1,3}\otimes dX_{2,4} + X_{1,4}\otimes dX_{2,3}
+ X_{3,4}\otimes dX_{1,2} \\
\end{cases}
\end{array}
\]
}
\end{Lemma}


\section{Periplectic Lie superalgebras and their desuperizations}


\ssec{The Lie superalgebras preserving symmetric odd bilinear forms
for $p>2$} We consider the following three types of periplectic
forms, i.e., symmetric odd bilinear forms with the Gram matrix
$\Pi_{n|n}$:

1) The Lie superalgebra $\fpe(n)$ that preserves this form,

2) $\fspe(n):=\fpe(n)\cap \fsle(n|n)$,

3) $\fp\fspe(pn):=\fspe(pn)/\fc$.

These three types of algebras, {\bf as well as the Lie superalgebras
of their outer derivations and central extensions} are ingredients
needed for the input of the generalized Cartan prolongations and
possible new simple Lie superalgebras (analogs of Lie algebras of
Hamiltonian vector fields) are concerned.

Recall that for $p=0$, $\fpe(n)$ has neither non-trivial extensions
nor outer derivatives; $\fspe(n)$ has only a central extension for
$n=4$, and the derivative
$d=\begin{pmatrix}1_n&0\\0&-1_n\end{pmatrix}$ for any $n>2$.

\sssbegin{Lemma} For $p>2$, we have $H^2(\fpe(n))=0$. Similarly,
$H^2(\fspe(n))=0$ for $n>4$ and $p>3$. Moreover, $H^2(\fspe(3))=0$
for $p>3$; however, for $p=3$ it is spanned by the following three
cocycles \footnotesize{\[
\begin{array}{ccl}
c_0^1&=&d(E^{2,3}-E^{6,5})\wedge d(E^{1,3}-E^{6,4})+d(E^{1,5}-E^{2,4})\wedge d(E^{6,3})\\[3mm]
c_{0}^2&=& d(E^{2,3}-E^{5,6})\wedge d(E^{1,2}-E^{5,4})+d(E^{1,6}-E^{3,4})\wedge d(E^{5,2})\\[3mm]
c_{0}^3&=& 2 d(E^{2,1}-E^{4,5})\wedge d(E^{3,1}-E^{4,6})+d(E^{2,6}-E^{3,5})\wedge d(E^{4,1})
\end{array}
\]
} In addition,  for every $p>2$, the space $H^2(\fspe(4))$ is
spanned by the cocycle
\[
c_{-2}=d(E^{1,6}-E^{2,5})\wedge d (E^{3,8}-E^{4,7})- d(E^{1,7}-E^{3,5})\wedge d (E^{2,8}-E^{4,6})+d (E^{1,8}-E^{4,5}) \wedge d (E^{2,7}-E^{3,8}).
\]
\end{Lemma}
\sssbegin{Lemma} For $p>2$, we have $H^1(\fpe(n);\fpe(n))=0$.
Moreover,  the space $H^1(\fspe(n);\fspe(n))$ is spanned by the
following derivative $d=\begin{pmatrix}1_n&0\\0&-1_n\end{pmatrix}$
for any $n>2$.
\end{Lemma}
\ssec{The Lie superalgebras preserving symmetric odd bilinear forms
for $p=2$ and their desuperizations}  For $p=2$, the analogs of
symplectic and periplectic Lie (super)algebras accrue additional
elements: If the matrix of the bilinear form $\cB$ is $\Pi_{2n}$
(resp. $\Pi_{n|n}$), then $\faut(\cB)$ consists of the
(super)matrices of the form
\begin{equation}
\label{matrgen}\begin{pmatrix} A & B\\C & A^t
\end{pmatrix},\end{equation}
where $B$ and $C$ are symmetric. Denote these {\it general} Lie
(super)algebras $\faut_{gen}(\cB)$; in particular, for
$\cB=\Pi_{2n}$ (resp. $\Pi_{n|n}$) we write $\fo_{gen}(2n)$ (resp.
$\fpe_{gen}(n)$). Clearly, $\fo_{gen}(2n)\simeq\fpe_{gen}(n)$ as
spaces.


Let $ZD$ denote the space of symmetric matrices with zeros on their
main diagonals. The derived Lie (super)algebra $\faut^{(1)}(\cB)$
consists of the (super)matrices of the form \eqref{matrgen}, where
$B,C\in ZD$.   In other words, these Lie (super)algebras resemble
the orthogonal Lie algebras. On these Lie (super)algebras
$\faut^{(1)}(\cB)$ the following (super)trace --- we call it the
\emph{half-trace} --- is defined:
\begin{equation}
\label{matrgen3} \htr: \begin{pmatrix} A & B\\C & A^t
\end{pmatrix}\tto \tr A.\end{equation}
The half-traceless Lie sub(super)algebra of $\faut^{(1)}(\cB)$ is
isomorphic to $\faut^{(2)}(\cB)$.

There is, however, a Lie superalgebra $\widetilde\faut(\cB)$, such
that $\faut^{(1)}(\cB)\subset\widetilde\faut(\cB)\subset\faut(\cB)$,
consisting of (super)matrices of the form \eqref{matrgen}, where
$C\in ZD$ while $B$ is just symmetric (obviously, there is an
isomorphic version of $\widetilde\faut(\cB)$ in which $B\in ZD$ and
$C$ is just symmetric). Denote this $\widetilde \faut(\cB)$ by
$\fo\fp(2n)$ if $\cB$ is even and $\fpe(n)$ if $\cB$ is odd.

We have, therefore, the following cases to consider assuming, as in
p.4, $\deg A^{i, i+1}=-\deg A^{i+1, i}=1$, and setting the degree of
the LOWEST (resp. HIGHEST) weight vector in the $\fgl(n)$-module of
matrices $B$ (resp. $C$) in \eqref{matrgen} be equal to 1 (resp.
$-1$):

1) $\fo_{gen}(2n)$ and $\fpe_{gen}(n)$ (in \eqref{matrgen}, both $B$
and $C$ symmetric);

1a) $\fp\fo_{gen}(2n):=\fo_{gen}(2n)/\Kee\  1_{2n}$ and
$\fp\fpe_{gen}(n):=\fpe_{gen}(n)/\Kee\  1_{2n}$;

2) $\fo\fp(2n)$ and $\fpe(n)$ (in \eqref{matrgen}, both $B$ and $C$
symmetric, but $C\in ZD$);

2a) $\fp\fo\fp(2n):=\fo_{gen}(2n)/\Kee\  1_{2n}$ and
$\fp\fpe(n):=\fpe_{gen}(n)/\Kee\  1_{2n}$;

2b)$\fo^{(1)}(2n)$  (in \eqref{matrgen}, both $B\in ZD$ and $C\in
ZD$);

2c)$\fp\fo^{(1)}(2n):=\fo^{(1)}(2n/\Kee\  1_{2n}$;

3) $\fs\fo\fp(2n)$ and $\fspe(n)$  (in \eqref{matrgen}, both $B$ and
$C$ symmetric, but $C\in ZD$, and $\tr A=0$);

3a) $\fp\fs\fo\fp(2n):=\fs\fo\fp(2n)/\Kee\   1_{2n}$ and
$\fp\fspe(n):=\fspe(n)/\Kee\  1_{2n}$;

4) $\fo^{(2)}(2n)$  (in \eqref{matrgen}, both $B\in ZD$ and $C\in
ZD$, and $\tr A=0$);

4a) $\fp\fo^{(2)}(2n):=\fo^{(2)}(2n)/\Kee\  1_{2n}$.

\sssbegin{Lemma} For $p=2$, the space $H^2(\fo_{gen}(2n))$ is
spanned by the following cocycles \footnotesize{
\begin{equation} \label{pogen}
\begin{array}{ll}
c_{-2}^{i,j}= d(E^{n+i,i})\wedge d(E^{n+j,j}),& 1\leq i< j \leq n\\[3mm]
c_{-1}^j=d(E^{j,j}+E^{n+j,n+j})\wedge d (E^{n+j,j})+&\\
\hskip 2cm \displaystyle\sum_{i\not = j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j}),&1\leq j \leq n\\[3mm]
c_0^{i,j}=d(E^{i,i+n})\wedge d(E^{n+j,j}),&1\leq i,j\leq n\text{~~except~}i=j=n. \\[3mm]
%\text{Degree 1}&:&c_{1}^{j}=d(E^{j,j}+E^{n+j,n+j})\wedge d (E^{j,n+j})+\displaystyle\sum_{i\not = j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i, n+j}+E^{j,n+i}), &1\leq j \leq n\\[3mm]
%\text{Degree 2}&:&c_2^{i,j}= d(E^{i,n+i})\wedge d(E^{j,j+n}) \quad 1 \leq i< j\leq n.
\end{array}
\end{equation}
} %The cocycle $c_0^{n,n}=d(E^{n,2n})\wedge d(E^{2n,n})$ is trivial.
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fpe_{gen}(2n))$ is
spanned by the cocycles \eqref{pogen} (except the cocycles
$c_{-1}^j$ and symmetric to them $c_1^j$ that become trivial), and
also the following cocycles:
\begin{equation}
\label{pegen}
\begin{array}{ll}
c_{-2}^i=  d(E^{n+i,i})^{\wedge^2},&1\leq i \leq n\\[2mm]
\tilde c_{-2}^{i,j}=  (d(E^{n+i,j} + E^{n+j,i}))^{\wedge^2},&1\leq i< j \leq n \\[2mm]
%\text{Degree 2}&:&c_2^{i}=  d(E^{i,n+i})^{\wedge^2} \quad 1\leq i \leq n\\[2mm]
%\tilde c_2^{i,j}&=&(d(E^{i,n+j}+E^{j,i+n}) )^{\wedge^2} \quad 1\leq j <i\leq n
\end{array}
\end{equation}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^2(\fo_{gen}(2n)/\Kee
1_{2n})$ is spanned by the following cocycles
\begin{equation}
\label{ogen2n}
\begin{array}{ll}
\text{$\deg=-2$}: &\text{Cocycles as in \eqref{pogen}}\\
\text{$\deg=-1$}: &\text{None}\\
\text{$\deg=0$}: &\text{Cocycles as in \eqref{pogen} and the cocycle } d(E^{n,2n})\wedge d (E^{2n,n})\\
%\text{Degree 1}: &=& \text{None}\\
%\text{Degree 2}: &=&\text{Cocycles as in (\ref{pogen})}\\
\end{array}
\end{equation}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fp\fe_{gen}(2n)/\Kee
1_{2n})$ is spanned by the following cocycles
\footnotesize{\begin{equation} \label{pegen2n}
\begin{array}{ll}
\text{$\deg=-2$}:&\text{Cocycles as in \eqref{pegen} and \eqref{pogen}}\\
\text{$\deg=-1$}:&\text{None}\\
\text{$\deg=0$}:&\text{Cocycles as in \eqref{pogen} and the cocycle }\\[2mm]
\text{for $n\neq 4$}:&
d(E^{n,2n})\wedge d (E^{2n,n})\\
\text{for $n= 4$}:&\displaystyle \sum_{i=2, 3,
4}d(E^{1,i}+E^{i+4,5})\wedge d(E^{i,1}+E^{5,i+4})
+\displaystyle \sum_{i=6, 7, 8}d(E^{1,i}+E^{i-4,5})\wedge d(E^{i,1}+E^{5,i-4}).\\[2mm]

%\text{Degree 1}:&=& \text{None}\\
%\text{Degree 2}:&=&\text{Cocycles as in (\ref{pogen}) and (\ref{pegen})}\\
\end{array}
\end{equation} }
\end{Lemma}

\sssbegin{Lemma} For $p=2$ and $\fg=\fo_{gen}(2n)$ or
$\fpe_{gen}(2n)$, the space $H^1(\fg;\fg)$ is spanned by the
following derivative $d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix}$ for
any $n>2$ and also the derivatives
\begin{equation} \label{opegen2n}
\begin{array}{ll}
c_{-1}^j=  \displaystyle \sum_{1\leq i\leq n}E^{i,i} \otimes d(E^{n+j,j}),&1\leq j \leq n\\[2mm]
%\text{Degree 1}&:&c_1^j=  \displaystyle\sum_{i=1}^n E^{i,i} \otimes d(E^{j,n+j})\quad 1\leq j \leq n
\end{array}
\end{equation}
\end{Lemma}

\sssbegin{Lemma} For $p=2$ and $\fg= \fo_{gen}(2n)/\Kee 1_{2n}$ or
$\fpe_{gen}(2n)/\Kee 1_{2n}$, , the space $H^1(\fg;\fg)$ is spanned
by the following derivative
$d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix}$ for any $n>2$.
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fo \fp(2n))$ is spanned
by the cocycles \footnotesize{\begin{equation} \label{op2n}
\begin{array}{lll}
\text{$\deg=-2$}:&\text{None}&\\
c_{-1}^{j}=& \displaystyle \sum_{i\not = j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j}),&1\leq j \leq n\\
\text{$\deg=0$}:& \text{None}&\\[2mm]
c_1^j=&\displaystyle \sum_{1\leq i\leq n} d(E^{i,i}+E^{i+n,i+n})\wedge d (E^{j,n+j}),&1\leq j \leq n\\[2mm]
\tilde c_1^j=&\displaystyle \sum_{i\not=j}
d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i,n+j}+E^{j,n+i})+
 d(E^{j,j}+E^{n+j,n+j})\wedge d (E^{j,n+j}),&1\leq j \leq n\\[2mm]
\text{$\deg=2$}:& \text{Cocycles as in \eqref{pogen}}&\\
\end{array}
\end{equation}
}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^2(\fpe(2n))$ is spanned by
the cocycles \footnotesize{\begin{equation} \label{pe2n}
\begin{array}{lll}
\text{$\deg=-2$}:&\tilde c_{-2}^{i,j}=  (d(E^{n+i,j} + E^{n+j,i}))^{\wedge^2},&1\leq i< j \leq n \\[2mm]
\text{$\deg=-1$}:& \text{None}&\\
\text{$\deg=0$}:& \text{None}&\\
\text{$\deg=1$}:&c_1^j=\displaystyle \sum_{i=1}^n d(E^{i,i}+E^{i+n,i+n})\wedge d (E^{j,n+j}),&1\leq j \leq n\\[2mm]
&\tilde c_1^j=\displaystyle \sum_{i\not=j}^n d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i,n+j}+E^{j,n+i})+ d(E^{j,j}+&\\
&E^{n+j,n+j})\wedge d (E^{j,n+j}),&1\leq j \leq n\\[2mm]
\text{$\deg=2$}:&c_{2}= \text{Cocycles as in \eqref{pogen} and \eqref{pegen}}&\\
\end{array}
\end{equation}
}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^1(\fo \fp (2n);\fo \fp
(2n))$ is spanned by the following derivatives\footnote{Hereafter
notation $\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix}$ and such like
designates a matrix with only one non-zero element, namely the block
represented by a matrix unit (here $E^{j,j}$).}
\footnotesize{\begin{equation} \label{op2n}
\begin{array}{ccl}
\text{$\deg=-1$}&:& \text{None}\\
\text{$\deg=0$}&:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$ and also the cocycle}\\[2mm]
&&c_0^2=
\displaystyle \sum_{1\leq i\leq 2n} E^{i,i}\otimes \left (d(E^{1,1}+E^{n+1,n+1})+\cdots + d(E^{n,n}+E^{2n,2n}) \right )\\[2mm]
\text{$\deg=1$}&:&c_1^j=\displaystyle \sum_{1\leq i\leq 2n} E^{i,i}d (E^{j,j+n}), \text{~~for~~} 1\leq j \leq n\text{ and a cocycle}\\[2mm]
&&\tilde c_1^j \text{ as multiplication by the matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$. }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fpe (2n);\fpe (2n))$ is
spanned by the following derivatives \footnotesize{\begin{equation}
\label{pe2n}
\begin{array}{ll}
\text{$\deg=-1$}:& \text{None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$ and also the cocycle}\\[2mm]
&c_0^2=
\displaystyle \sum_{1\leq i\leq 2n} E^{i,i}\otimes \left (d(E^{1,1}+E^{n+1,n+1})+\cdots + d(E^{n,n}+E^{2n,2n}) \right )\\[2mm]
\text{$\deg=1$}:& \tilde c_1^j \text{ as multiplication by the
matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$. }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fp \fo \fp(2n))$ is
spanned by the following cocycles \footnotesize{\begin{equation}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None}\\
\text{$\deg=-1$}:& \text{None}\\
\text{$\deg=0$}:&\text{None for $n\not =4$. For $n=4$, the cocycle is:}  \\
c_0=&d(E^{1,2}+E^{6,5}) \wedge d(E^{2,1}-E^{5,6})+d(E^{2,3}+E^{7,6}) \wedge d(E^{3,2}-E^{6,7})\\[2mm]
&+ d(E^{2,4}+E^{8,6}) \wedge d(E^{4,2}-E^{6,8})+
d(E^{2,7}+E^{3,6}) \wedge d(E^{7,2}-E^{6,3})\\[2mm]
&+d(E^{2,8}+E^{4,6}) \wedge d(E^{6,4}-E^{8,2})+
d(E^{1,6}+E^{2,5}) \wedge d(E^{5,2}-E^{6,1}) \\[2mm]
c_{1}^j=&\displaystyle \sum_{2\leq i\leq n} d(E^{i,i}+E^{i+n,i+n})\wedge d (E^{j,j+n}),\text{~where~} 1\leq j \leq n. \\
\text{$\deg=2$}:&\text{Cocycles as in \eqref{pogen}}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fp \fo \fp (2n);\fp \fo
\fp (2n))$ is spanned by the following derivatives
\footnotesize{\begin{equation} \label{pop2n}
\begin{array}{ll}
\text{$\deg=-1$}:& \text{None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
\text{$\deg=1$}:& \tilde c_1^j \text{ as multiplication by the
matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^2(\fp \fp \fe(2n))$ is
spanned by the following cocycles:
\footnotesize{\begin{equation}\label{ppe2n}
\begin{array}{ll}
\text{$\deg=-2$}:&c_{-2}^{i,j}=  (d(E^{n+i,j} + E^{n+j,i}))^{\wedge^2},\text{~~where ~~}1\leq i< j \leq n \\[2mm]
\text{$\deg=-1$}:& \text{ None }\\
\text{$\deg=0$}:&\text{None for $n\not =4$. For $n=4$, the cocycle is }  \\
c_0= &d(E^{1,2}+E^{6,5}) \wedge d(E^{2,1}+E^{5,6})+d(E^{2,3}+E^{7,6}) \wedge d(E^{3,2}+E^{6,7})\\[2mm]
&+ d(E^{2,4}+E^{8,6}) \wedge d(E^{4,2}+E^{6,8})+
d(E^{2,7}+E^{3,6}) \wedge d(E^{7,2}+E^{6,3})\\[2mm]
&+d(E^{2,8}+E^{4,6}) \wedge d(E^{6,4}+E^{8,2})+
d(E^{1,6}+E^{2,5}) \wedge d(E^{5,2}+E^{6,1}) \\[2mm]
\text{$\deg=1$}:&\displaystyle \sum_{2\leq i\leq n} d(E^{i,i}+E^{i+n,i+n})\wedge d (E^{j,j+n}),\text{~~where ~~} 1\leq j \leq n. \\[2mm]
\text{$\deg=2$}:& \text{Cocycles as in \eqref{pogen} and
\eqref{pegen} }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$ and $\fg=\fp \fo \fp (2n)$ or $\fp \fp
\fe (2n)$, the space $H^1(\fg;\fg)$ is spanned by the following
derivatives \footnotesize{\begin{equation} \label{ppe2n}
\begin{array}{ll}
\text{$\deg=-1$}:& \text{None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
\text{$\deg=1$}:& \tilde c_1^j \text{ as multiplication by the
matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}




\sssbegin{Lemma} For $p=2$, the space $H^2(\fo^{(1)}(2n))$ is
spanned by the following cocycles:
\footnotesize{\begin{equation}\label{o1(2n)}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None}\\[3mm]
c_{-1}^j=&\displaystyle\sum_{i\not = j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j}), \quad 1\leq j \leq n\\[3mm]
c_0=&\displaystyle \sum_{2\leq i\leq n} d(E^{1,i}+E^{i+n,n+1})\wedge d (E^{i,1}+E^{n+1,i+n})\\
&\displaystyle+ \sum_{2 \leq i < j \leq n} d(E^{i,n+j}+E^{j,n+i})\wedge d (E^{n+j,i}+E^{n+i,j})\\[3mm]
%\text{$\deg=1$}:&c_{1}^{j}=\displaystyle\sum_{i\not = j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i, n+j}+E^{j,n+i}), \quad 1\leq j \leq n\\[3mm]
%\text{$\deg=2$}:&\text{None}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fo^{(1)} (2n);\fo^{(1)}
(2n))$ is spanned by the following derivatives
\footnotesize{\begin{equation} \label{o12n}
\begin{array}{ll}
\text{$\deg=-1$}:&\tilde c_{-1}^j \text{ as multiplication by the
matrix }
 d_j=\begin{pmatrix}0&E^{j,j}\\0&0\end{pmatrix} \text{ for any $n>2$ }\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$ and also the cocycle}\\[2mm]
&c_0^2=
\displaystyle \sum_{1\leq i\leq 2n} E^{i,i}\otimes \left (d(E^{1,1}+E^{n+1,n+1})+\cdots + d(E^{n,n}+E^{2n,2n}) \right )\\[2mm]
\text{$\deg=1$}:& \tilde c_1^j \text{ as multiplication by the
matrix }
 d_j^t=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^2(\fp \fo^{(1)}(2n))$ is
spanned by the following  cocycles:
\footnotesize{\begin{equation}\label{po1(2n)}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None}\\[3mm]
c_{-1}^j=&\displaystyle\sum_{i\not = j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j}), \quad 1\leq j \leq n\\[3mm]
c_0=&\displaystyle \sum_{2\leq i\leq n} d(E^{1,i}+E^{i+n,n+1})\wedge d (E^{i,1}+E^{n+1,i+n})%\\[3mm]
\displaystyle+ \sum_{2 \leq i < j \leq n} d(E^{i,n+j}+E^{j,n+i})\wedge d (E^{n+j,i}+E^{n+i,j})\\[3mm]
& \text{For $n=4$, we have one more cocycle }\\[3mm]
\tilde c_0=& d(E^{1,3}+E^{7,5})\wedge d(E^{3,1}+E^{5,7})+d(E^{1,4}+
E^{8,5})\wedge d(E^{4,1}+E^{5,8})\\
&+ d(E^{2,3}+E^{7,6})\wedge d (E^{3,2}+E^{6,7})+d(E^{2,4}+E^{8,6})\wedge d(E^{4,2}+
E^{6,8})\\[3mm]
&+d(E^{3,8}+E^{4,7})\wedge d (E^{7,4}+E^{8,3})+d(E^{1,6}+E^{2,5})\wedge d(E^{5,2}+
E^{6,1})\\[3mm]
%\text{$\deg=1$}:&c_{1}^{j}=\displaystyle\sum_{i\not = j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i, n+j}+E^{j,n+i}), \quad 1\leq j \leq n\\[3mm]
%\text{$\deg=2$}:&\text{None}\\
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fg;\fg)$, for $\fg=\fp
\fo^{(1)} (2n)$, is spanned by the following derivatives
\footnotesize{\begin{equation} \label{po1(2n)}
\begin{array}{ll}
\text{$\deg=-1$}:&\tilde c_{-1}^j \text{ as multiplication by the
matrix }
 d_j=\begin{pmatrix}0&E^{j,j}\\0&0\end{pmatrix} \text{ for any $n>2$ }\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
\text{$\deg=1$}:& \tilde c_1^j \text{ as multiplication by the
matrix }
 d_j*t=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fs \fo \fp(2n))$ is
spanned by the following cocycles:
\footnotesize{\begin{equation}\label{sop2n}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None for $n\neq 4$. For $n=4$, we have the cocycle}\\[3mm]
c_{-2}=&d(E^{7,4}+E^{8,3})\wedge d (E^{5,2}+E^{6,1})+ d (E^{6,3}+E^{7,2})\wedge d  (E^{5,4}+E^{8,1})\\
&+d(E^{6,4}+E^{8,2})\wedge d (E^{5,3}+E^{7,1})\\[3mm]
%\text{$\deg=-1$}:&\\[3mm]
c_{-1}^{j}=& \displaystyle \sum_{i\not = j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j})\quad 1\leq j \leq n\\
\text{$\deg=0$}:& \text{None }\\[3mm]
c_{1}^{j}=&\displaystyle\sum_{i\not = j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i, n+j}+E^{j,n+i})\\[4mm]
& + \left \{
\begin{array}{ll}
d(E^{1,1}+E^{2,2}+E^{1+n,1+n}+E^{2+n,2+n})\wedge d (E^{1,n})& \text{for } i=1\\
d(E^{n-1,n-1}+E^{n,n}+E^{2n-1,2n-1}+E^{2n,2n})\wedge d (E^{n,2n})& \text{for } i=n\\
d(E^{j-1,j-1}+E^{j+1,j+1}+E^{j-1+n,j-1+n}+E^{j+1+n,j+1+n})\wedge d (E^{j,j+n})& \text{for } j\not=1,n\\
\end{array}
\right.\\[3mm]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen}}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fs \fo \fp (2n);\fs \fo
\fp (2n))$ is spanned by the following derivatives
\footnotesize{\begin{equation} \label{sop(2n)}
\begin{array}{ll}
\text{$\deg=-1$}:&\text{ None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
& \text{If $n$ is even, we have the additional cocycle as
multiplication by the matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
\text{$\deg=1$}:&  c_1^j \text{ as multiplication by the matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fs \fo \fp(4n)/\Kee
1_{4n})$ is spanned by the  following cocycles:
\footnotesize{\begin{equation}\label{sop4n}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None for $n\neq 2$. For $n=2$, we have the cocycle}\\[3mm]
c_{-2}=& d(E^{7,4}+E^{8,3})\wedge d (E^{5,2}+E^{6,1})+ d (E^{6,3}+E^{7,2})\wedge d  (E^{5,4}+E^{8,1})\\
&+d(E^{6,4}+E^{8,2})\wedge d (E^{5,3}+E^{7,1})\\[3mm]
%\text{$\deg=-1$}:&\\[3mm]
c_{-1}^{j}=& \displaystyle \sum_{i\not = j} d(E^{i,j}+E^{j+2n,i+2n})\wedge d (E^{j+2n,i}+E^{i+2n,j})\quad 1\leq j \leq 2n\\
c_0=& \displaystyle \sum_{1\leq i\leq 2n} d(E^{1,i}+E^{2n+i,2n+1})\wedge d (E^{i,1}+E^{2n+1,2n+i})\\[3mm]
\text{$\deg=1$}:&\text{None }\\[3mm]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen}}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$  and $\fg=\fs \fo \fp (4n)/\Kee 1_{4n}$,
the space $H^1(\fg;\fg)$ is spanned by the following derivatives
\footnotesize{\begin{equation} \label{sop(4n)/c}
\begin{array}{ll}
\text{$\deg=-2$}:&\text{ None for $n\neq 2$. For $n=2$,  we have the cocycle}\\
c _{-2}=&(E^{3,8}+E^{4,7})\otimes d (E^{5,2}+E^{6,1})+(E^{2,7}+E^{3,6})\otimes d (E^{5,4}+E^{8,1})\\[3mm]
&+(E^{2,8}+E^{4,6})\otimes d (E^{5,3}+E^{7,1})+(E^{1,6}+E^{2,5})\otimes d (E^{7,4}+E^{8,3})\\[3mm]
&+(E^{1,7}+E^{3,5})\otimes d (E^{6,4}+E^{8,2})+(E^{1,8}+E^{4,5})\otimes d (E^{6,3}+E^{7,2})\\[3mm]
\text{$\deg=-1$}:&\text{ None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_{2n}\end{pmatrix} \text{ for any $n>1$}\\[2mm]
& \text{and also the cocycle given as multiplication by the matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
\text{$\deg=1$}:& \text{ None}
\end{array}
\end{equation}
}
\end{Lemma}


\sssbegin{Lemma} For $p=2$, the space $H^2(\fs \fp \fe(2n))$ is
spanned by the following cocycles:
\footnotesize{\begin{equation}\label{spe2n}
\begin{array}{ll}
\text{$\deg=-2$}:& \tilde c_{-2}^{i,j}=(d(E^{n+i,j}+E^{n+j,i}) )^{\wedge^2} \quad 1\leq i <j\leq n \\[3mm]
& \text{For $n=4$, we have one more cocycle}\\
& c_{-2}=d(E^{7,4}+E^{8,3})\wedge d (E^{5,2}+E^{6,1})+ d (E^{6,3}+E^{7,2})\wedge d  (E^{5,4}+E^{8,1})\\
&+d(E^{6,4}+E^{8,2})\wedge d (E^{5,3}+E^{7,1})\\[3mm]
\text{$\deg=-1$}:& \text{ None}\\[3mm]
\text{$\deg=0$}:& \text{None }\\[3mm]
\text{$\deg=1$}:&c_{1}^{j}=\displaystyle\sum_{i\not = j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i, n+j}+E^{j,n+i})\\[4mm]
& + \left \{
\begin{array}{ll}
d(E^{1,1}+E^{2,2}+E^{1+n,1+n}+E^{2+n,2+n})\wedge d (E^{1,n})& \text{for } i=1\\
d(E^{n-1,n-1}+E^{n,n}+E^{2n-1,2n-1}+E^{2n,2n})\wedge d (E^{n,2n})& \text{for } i=n\\
d(E^{j-1,j-1}+E^{j+1,j+1}+E^{j-1+n,j-1+n}+E^{j+1+n,j+1+n})\wedge d (E^{j,j+n})& \text{for } j\not=1,n\\
\end{array}
\right.\\[3mm]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen} and
\eqref{pegen}}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$  and $\fg=\fs \fp \fe (2n)$, the space
$H^1(\fg;\fg)$ is spanned by the following derivatives
\footnotesize{\begin{equation} \label{spe(2n)}
\begin{array}{ll}
\text{$\deg=-1$}:&\text{ None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
& \text{If $n$ is even, we have a cocycle as multiplication by the
matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
\text{$\deg=1$}:&  c_1^j \text{ as multiplication by the matrix }
 d_j=\begin{pmatrix}0&0\\E^{j,j}&0\end{pmatrix} \text{ for any $n>2$ }
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fs \fp \fe(4n)/ \Kee
1_{4n})$ is spanned by the following cocycles:
\footnotesize{\begin{equation}\label{spe4n/c}
\begin{array}{ll}
\text{$\deg=-2$}:& \tilde c_{-2}^{i,j}=(d(E^{2n+i,j}+E^{2n+j,i}) )^{\wedge^2} \quad 1\leq i <j\leq 2n \\[3mm]
& \text{For $n=2$, we have one more cocycle}\\
c_{-2}=&d(E^{7,4}+E^{8,3})\wedge d (E^{5,2}+E^{6,1})+ d (E^{6,3}+E^{7,2})\wedge d  (E^{5,4}+E^{8,1})\\
&+d(E^{6,4}+E^{8,2})\wedge d (E^{5,3}+E^{7,1})\\[3mm]
\text{$\deg=-1$}:& \text{ None}\\[3mm]
c_0=&\displaystyle \sum_{1\leq i\leq 2n} d(E^{1,i}+E^{2n+i,2n+1})\wedge d (E^{i,1}+E^{2n+1,2n+i})\\[3mm]
\text{$\deg=1$}:& \text{ None}\\[3mm]
\text{$\deg=2$}:&\text{ cocycles as in \eqref{pogen} and
\eqref{pegen}}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$ and $\fg=\fs \fp \fe (4n)/\Kee 1_{4n}$,
the space $H^1(\fg;\fg)$  is spanned by the following derivatives
\footnotesize{\begin{equation} \label{spe(4n)/c}
\begin{array}{ll}
\text{$\deg=-2$}:&\text{ None for $n\neq 2$. For $n=2$, we have the cocycle}\\
c _{-2}=&(E^{3,8}+E^{4,7})\otimes d (E^{5,2}+E^{6,1})+(E^{2,7}+E^{3,6})\otimes d (E^{5,4}+E^{8,1})\\[3mm]
&+(E^{2,8}+E^{4,6})\otimes d (E^{5,3}+E^{7,1})+(E^{1,6}+E^{2,5})\otimes d (E^{7,4}+E^{8,3})\\[3mm]
&+(E^{1,7}+E^{3,5})\otimes d (E^{6,4}+E^{8,2})+(E^{1,8}+E^{4,5})\otimes d (E^{6,3}+E^{7,2})\\[3mm]
\text{$\deg=-1$}:&\text{ None}\\
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_{2n}\end{pmatrix} \text{ for any $n>1$}\\[2mm]
& \text{and another cocycle as multiplication by the matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
\text{$\deg=1$}:& \text{ None}
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fo^{(2)}(2n))$ is
spanned by the following cocycles:
\tiny{\begin{equation}\label{o2(2n)}
\begin{array}{ll}
\text{$\deg=-2$}:& \text{None for $n\neq 4$. For $n=4$,  we have the cocycle}\\
c_{-2}=&d(E^{7,4}+E^{8,3})\wedge d (E^{5,2}+E^{6,1})+ d (E^{6,3}+E^{7,2})\wedge d  (E^{5,4}+E^{8,1})\\
&+d(E^{6,4}+E^{8,2})\wedge d (E^{5,3}+E^{7,1})\\[3mm]
\text{$\deg=-1$}:& c^j_{-1}=\displaystyle \sum_{i\not =j} d(E^{i,j}+E^{j+n,i+n})\wedge d (E^{j+n,i}+E^{i+n,j})\quad 1 \leq j \leq n\\[3mm]
& \text{For $n=4$, there are also the following $4$ cocycles:}\\
&c_{-1}^1=d(E^{1,2}+E^{6,5})\wedge d (E^{7,4}+E^{8,3})+ d (E^{1,3}+E^{7,5})\wedge d(E^{6,4}+E^{8,2})\\
&+d(E^{1,4}+E^{8,5})\wedge d (E^{6,3}+E^{7,2})\\[3mm]
&c_{-1}^2=d(E^{2,3}+E^{7,6})\wedge d (E^{5,4}+E^{8,1})+ d (E^{2,4}+E^{8,6})\wedge d(E^{5,3}+E^{7,1})\\
&+d(E^{2,1}+E^{5,6})\wedge d (E^{7,4}+E^{8,3})\\[3mm]
&c_{-1}^3=d(E^{3,4}+E^{8,7})\wedge d (E^{5,2}+E^{6,1})+ d (E^{3,2}+E^{6,7})\wedge d(E^{5,4}+E^{8,1})\\
&+d(E^{3,1}+E^{5,7})\wedge d (E^{6,4}+E^{8,2})\\[3mm]
&c_{-1}^4=d(E^{4,3}+E^{7,8})\wedge d (E^{5,2}+E^{6,1})+ d (E^{4,2}+E^{6,8})\wedge d(E^{5,3}+E^{7,1})\\
&+d(E^{4,1}+E^{5,8})\wedge d (E^{6,3}+E^{7,2})\\[3mm]

\text{$\deg=0$}:& c_0=\displaystyle \sum_{1\leq i\leq n} d(E^{1,i}+E^{n+i,n+1})\wedge d (E^{i,1}+E^{n+1,n+i})\\[3mm]
&+ \displaystyle \sum_{2\leq i <j\leq n} d(E^{i,n+j}+E^{j,n+i})\wedge d (E^{n+j,i}+E^{n+i,j})\\[3mm]
& \text{For $n=4$, there are also six cocycles, given by}\\
&c_0^1=d(E^{3,2}-E^{6,7})\wedge
   d(E^{4,1}-E^{5,8}) +d(E^{4,2}-E^{6,8})\wedge
   d(E^{3,1}-E^{5,7})\\
&
   +d(E^{3,8}-E^{4,7})\wedge
   d(E^{5,2}+E^{6,1})\\[3mm]
& c_0^2=d(E^{3,4}-E^{8,7})\wedge
   d(E^{2,1}-E^{5,6}) +d(E^{2,4}-E^{8,6})\wedge
   d(E^{3,1}-E^{5,7})\\
&+d(E^{2,7}-E^{3,6})\wedge
   d(E^{5,4}+E^{8,1})\\[3mm]
& c_0^3= d(E^{4,3}-E^{7,8})\wedge
   d(E^{2,1}-E^{5,6})+
d(E^{2,3}-E^{7,6})\wedge
   d(E^{4,1}-E^{5,8}) \\
& +d(E^{2,8}-E^{4,6})\wedge
   d(E^{5,3}+E^{7,1})\\[3mm]
&c_0^4=d(E^{2,3}-E^{7,6})\wedge
   d(E^{1,4}-E^{8,5}) +d(E^{2,4}-E^{8,6})\wedge
   d(E^{1,3}-E^{7,5})\\
&+d(E^{1,6}-E^{2,5})\wedge
   d(E^{7,4}+E^{8,3})\\
&c_0^5=d(E^{3,4}-E^{8,7})\wedge
   d(E^{1,2}-E^{6,5})+
d(E^{3,2}-E^{6,7})\wedge
   d(E^{1,4}-E^{8,5})\\
&+ d(E^{1,7}-E^{3,5})\wedge
   d(E^{6,4}+E^{8,2})\\[3mm]
&c_0^6=d(E^{4,3}-E^{7,8})\wedge
   d(E^{1,2}-E^{6,5})+d(E^{4,2}-E^{6,8})\wedge
   d(E^{1,3}-E^{7,5})\\
&+ d(E^{1,8}-E^{4,5})\wedge
   d(E^{6,3}+E^{7,2})\\[3mm]
\text{$\deg=1$}:& c^j_{1}=\displaystyle \sum_{i\not =j} d(E^{j,i}+E^{i+n,j+n})\wedge d (E^{i,j+n}+E^{j,i+n})\quad 1 \leq j \leq n\\[3mm]
& \text{For $n=4$, there also the following cocycles:}\\
&c_{1}^1=d(E^{2,1}+E^{5,6})\wedge d (E^{4,7}+E^{3,8})+ d (E^{3,1}+E^{5,7})\wedge d(E^{4,6}+E^{2,8})\\
&+d(E^{4,1}+E^{5,8})\wedge d (E^{3,6}+E^{2,7})\\[3mm]
&c_{1}^2=d(E^{3,2}+E^{6,7})\wedge d (E^{4,5}+E^{1,8})+ d (E^{4,2}+E^{6,8})\wedge d(E^{3,5}+E^{1,7})\\
&+d(E^{1,2}+E^{6,5})\wedge d (E^{4,7}+E^{3,8})\\[3mm]
&c_{1}^3=d(E^{4,3}+E^{7,8})\wedge d (E^{2,5}+E^{1,6})+ d (E^{2,3}+E^{7,6})\wedge d(E^{4,5}+E^{1,8})\\
&+d(E^{1,3}+E^{7,5})\wedge d (E^{4,6}+E^{2,8})\\[3mm]
&c_{1}^4=d(E^{3,4}+E^{8,7})\wedge d (E^{2,5}+E^{1,6})+ d (E^{2,4}+E^{8,6})\wedge d(E^{3,5}+E^{1,7})\\
&+d(E^{1,4}+E^{8,5})\wedge d (E^{3,6}+E^{2,7})\\[3mm]
\text{$\deg=2$}:& \text{None, except for $n=4$ we have the cocycle}\\
& c_{2}=d(E^{4,7}+E^{3,8})\wedge d (E^{2,5}+E^{1,6})+ d (E^{3,6}+E^{2,7})\wedge d  (E^{4,5}+E^{1,8})\\
&+d(E^{4,6}+E^{2,8})\wedge d (E^{3,5}+E^{1,7})\\[3mm]
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^1(\fg;\fg)$, for
$\fg=\fo^{(2)}(2n)$, is spanned by the following derivatives
\footnotesize{\begin{equation} \label{o2(2n)}
\begin{array}{ll}
\text{$\deg=-1$}:&\text{as multiplication by the matrix }
 d_i=\begin{pmatrix}0&E^{i,i}\\0&0\end{pmatrix} \text{ for any $n>2$ } (i=1,\ldots,n)\\[2mm]
\text{$\deg=0$}:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>2$}\\[2mm]
& \text{If $n$ is even, there is one more
cocycle as multiplication by the matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
\text{$\deg=1$}:& \text{as multiplication by the matrix }
 d_i^t=\begin{pmatrix}0&0\\E^{i,i}&0\end{pmatrix} \text{ for any $n>2$ }  (i=1,\ldots,n)\\[2mm]
\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$, the space $H^2(\fo^{(2)}(4n)/\Kee
1_{4n})$ is spanned by the   following cocycles:
\footnotesize{\begin{equation}\label{o2(4n)/c}
\begin{array}{ccl}
\text{$\deg=-2$}&:& \text{As in the case of $\fo^{(2)}(4n)$}\\
\text{$\deg=-1$}&:& \text{As in the case of $\fo^{(2)}(4n)$}\\
\text{$\deg=0$}&:& \text{As in the case of $\fo^{(2)}(4n)$ and one more cocycle}\\
&&\tilde c_0=\displaystyle \sum_{1\leq i\leq n}
d(E^{1,i}+E^{i+n,n+1})\wedge
   d(E^{i,1}+E^{n+1,n+i})\\
\text{$\deg=1$}&:& \text{As in the case of $\fo^{(2)}(4n)$}\\
\text{$\deg=2$}&:& \text{As in the case of $\fo^{(2)}(4n)$}\\

\end{array}
\end{equation}
}
\end{Lemma}

\sssbegin{Lemma} For $p=2$ and $\fg=\fo^{(2)}(4n)/\Kee 1_{4n}$,  the
space $H^1(\fg;\fg)$ is spanned by the following derivatives
\footnotesize{\begin{equation}\label{dero2(4n)/c}
\begin{array}{ccl}
\text{$\deg=-2$}&:&
   E^{3,8} \otimes d (E^{5,2}+E^{6,1} )-E^{4,7} \otimes d (E^{5,2}+E^{6,1} )+E^{2,7} \otimes d (E^{5,4}+E^{8,1}  )\\
&&-E^{3,6} \otimes d (E^{5,4}+E^{8,1} )+E^{2,8} \otimes d (E^{5,3}+E^{7,1} )-E^{4,6}\otimes  d (E^{5,3}+E^{7,1})+\\
&&E^{1,6}\otimes (d(E^{7,4}+E^{8,3})-E^{2,5}\otimes d
   (E^{7,4}+E^{8,3})+E^{1,7}\otimes d(E^{6,4}+E^{8,2})\\
&&-E^{3,5} \otimes d (E^{6,4}+E^{8,2} )+E^{1,8}\otimes  d(E^{6,3}+E^{7,2})-E^{4,5}\otimes d(E^{6,3}+E^{7,2})\\[3mm]
\text{$\deg=-1$}&:&\text{as multiplication by the matrix }
 d_i=\begin{pmatrix}0&E^{i,i}\\0&0\end{pmatrix} \text{ for any $n>2$ } (i=1,\ldots,n)\\[2mm]
&&\text{For $n=2$ there are also four cocycles given by}\\
&&c_{-1}^1=E^{2,1}\otimes d(E^{7,4}+E^{8,3})
+E^{5,6}\otimes d (E^{7,4}+E^{8,3} )+
   E^{3,1}\otimes d(E^{6,4}+E^{8,2}) \\
&& +E^{5,7}\otimes d (E^{6,4}+E^{8,2} ) +
   E^{4,1}\otimes d(E^{6,3}+E^{7,2})
  +E^{5,8}\otimes d (E^{6,3}+E^{7,2} )\\
&& +
   E^{3,8}\otimes d(E^{1,2}-E^{6,5})
 +E^{4,7}\otimes d (E^{1,2}-E^{6,5} ) +
   E^{2,7}\otimes d(E^{1,4}-E^{8,5})\\
&&
+E^{3,6}\otimes d(E^{1,4}-E^{8,5})+
   E^{2,8}\otimes d(E^{1,3}-E^{7,5})
   +E^{4,6}\otimes d (E^{1,3}-E^{7,5} )\\[3mm]
&&c_{-1}^2= E^{1,4}\otimes d (E^{6,3}+E^{7,2} )
  +E^{8,5}\otimes d (E^{6,3}+E^{7,2} ) +
   E^{2,4}\otimes d (E^{5,3}+E^{7,1} )\\
&&
+E^{8,6}\otimes d (E^{5,3}+E^{7,1} ) +
  E^{3,4}\otimes d (E^{5,2}+E^{6,1} )
 +E^{8,7}\otimes d (E^{5,2}+E^{6,1} ) \\
&&+
 E^{2,7}\otimes d (E^{4,1}-E^{5,8} )+
E^{3,6}\otimes d (E^{4,1}-E^{5,8} ) +
 E^{1,6}\otimes d (E^{4,3}-E^{7,8} ) \\
&&
 +E^{2,5}\otimes d (E^{4,3}-E^{7,8} ) +
 E^{1,7}\otimes d (E^{4,2}-E^{6,8} )
+E^{3,5}\otimes d (E^{4,2}-E^{6,8} ) \\[3mm]
&& c_{-1}^3= E^{1,2}\otimes d (E^{7,4}+E^{8,3} )
   +E^{6,5}\otimes d (E^{7,4}+E^{8,3} ) +
 E^{3,2}\otimes d (E^{5,4}+E^{8,1} )\\
&&
+E^{6,7}\otimes d (E^{5,4}+E^{8,1} ) +
   E^{4,2}\otimes d (E^{5,3}+E^{7,1} )+
E^{6,8}\otimes d (E^{5,3}+E^{7,1} ) \\
&&+
 E^{3,8}\otimes d (E^{2,1}-E^{5,6} )+
E^{4,7}\otimes d (E^{2,1}-E^{5,6} ) +
  E^{1,7}\otimes d (E^{2,4}-E^{8,6} )\\
&&+E^{3,5}\otimes d (E^{2,4}-E^{8,6} ) +
 E^{1,8}\otimes d (E^{2,3}-E^{7,6} )+
E^{4,5}\otimes d (E^{2,3}-E^{7,6} )\\[3mm]
&&c_{-1}^4= E^{1,3}\otimes d (E^{6,4}+E^{8,2} )+E^{7,5}\otimes d (E^{6,4}+E^{8,2} ) +
   E^{2,3}\otimes d (E^{5,4}+E^{8,1} )\\
&&+E^{7,6}\otimes d (E^{5,4}+E^{8,1} ) +
 E^{4,3}\otimes d (E^{5,2}+E^{6,1} )+
 E^{7,8}\otimes d (E^{5,2}+E^{6,1})\\
&& +
  E^{2,8}\otimes d (E^{3,1}-E^{5,7} )
+E^{4,6}\otimes d (E^{3,1}-E^{5,7} ) +
 E^{1,6}\otimes d (E^{3,4}-E^{8,7} ) \\
&&+E^{2,5}\otimes d (E^{3,4}-E^{8,7} )+
   E^{1,8}\otimes d (E^{3,2}-E^{6,7} )
+E^{4,5}\otimes d (E^{3,2}-E^{6,7} )\\[3mm]
\text{$\deg=0$}&:& \text{as multiplication by the matrix }
 d=\begin{pmatrix}0&0\\0&1_n\end{pmatrix} \text{ for any $n>1$}\\[2mm]
&& \text{and the cocycle as multiplication by the matrix }
 d_{1,1}=\begin{pmatrix}E^{1,1}&0\\0&E^{1,1}\end{pmatrix}\\[2mm]
&&\text{If $n=2$, there are six cocycles given by}\\[2mm]
&&c_0^1= E^{3,1}\otimes d (E^{2,4}-E^{8,6} )
+E^{5,7}\otimes d (E^{2,4}-E^{8,6} ) +
  E^{4,1}\otimes d (E^{2,3}-E^{7,6} )\\
&&+E^{5,8}\otimes d (E^{2,3}-E^{7,6} ) +
  E^{3,2}\otimes d (E^{1,4}-E^{8,5} )
 +E^{6,7}\otimes d (E^{1,4}-E^{8,5} ) \\
&&+
  E^{4,2}\otimes d (E^{1,3}-E^{7,5} )
+E^{6,8}\otimes d (E^{1,3}-E^{7,5} )
+E^{3,8}\otimes d (E^{1,6}-E^{2,5})\\
&&+E^{4,7}\otimes d (E^{1,6}-E^{2,5} )  +
  +E^{5,2}\otimes d (E^{7,4}+E^{8,3} )  +
 E^{6,1}\otimes d (E^{7,4}+E^{8,3} )\\[3mm]
&&c_0^2= (E^{2,4}\otimes d (E^{1,3}-E^{7,5} )
   +E^{8,6}\otimes d (E^{1,3}-E^{7,5} )  +
  E^{3,4}\otimes d (E^{1,2}-E^{6,5} )\\
&&
+E^{8,7}\otimes d (E^{1,2}-E^{6,5} ) +
   E^{2,1}\otimes d (E^{4,3}-E^{7,8} )
 +E^{5,6}\otimes d (E^{4,3}-E^{7,8} ) \\
&& +
   E^{3,1}\otimes d (E^{4,2}-E^{6,8} )
+E^{5,7}\otimes d (E^{4,2}-E^{6,8} ) +
  E^{2,7}\otimes d (E^{1,8}-E^{4,5} )\\
&&
+E^{3,6}\otimes d (E^{1,8}-E^{4,5} ) +
   E^{5,4}\otimes d (E^{6,3}+E^{7,2} )  +
  E^{8,1}\otimes d (E^{6,3}+E^{7,2} )\\[3mm]
&&c_0^3= E^{2,3}\otimes d (E^{1,4}-E^{8,5} )
  +E^{7,6}\otimes d (E^{1,4}-E^{8,5} ) +
  E^{2,1}\otimes d (E^{3,4}-E^{8,7} )\\
&&
 +E^{5,6}\otimes d (E^{3,4}-E^{8,7} ) +
   E^{4,1}\otimes d (E^{3,2}-E^{6,7} )
 +E^{5,8}\otimes d (E^{3,2}-E^{6,7} ) \\
&&+
   E^{4,3}\otimes d (E^{1,2}-E^{6,5} )
 +E^{7,8}\otimes d (E^{1,2}-E^{6,5} ) +
   E^{2,8}\otimes d (E^{1,7}-E^{3,5} )\\
&&
  +E^{4,6}\otimes d (E^{1,7}-E^{3,5} ) +
   E^{5,3}\otimes d (E^{6,4}+E^{8,2} +
 E^{7,1}\otimes d (E^{6,4}+E^{8,2} )\\[3mm]
\end{array}
\end{equation}}
\footnotesize{\begin{equation}\label{dero2(4n)/c}
\begin{array}{lcl}
&&c_0^4=(E^{1,3}\otimes d (E^{4,2}-E^{6,8} )
 +E^{7,5}\otimes d(E^{4,2}-E^{6,8} ) +
 E^{1,4}\otimes d (E^{3,2}-E^{6,7} )\\
&&
 +E^{8,5}\otimes d (E^{3,2}-E^{6,7} ) +
   E^{2,3}\otimes d (E^{4,1}-E^{5,8} )
  +E^{7,6}\otimes d (E^{4,1}-E^{5,8} ) \\
&&+
    E^{2,4}\otimes d (E^{3,1}-E^{5,7} )
+E^{8,6}\otimes d (E^{3,1}-E^{5,7} ) +
  E^{1,6}\otimes d (E^{3,8}-E^{4,7} )\\
&&
 +E^{2,5}\otimes d (E^{3,8}-E^{4,7} ) +
   E^{7,4}\otimes d (E^{5,2}+E^{6,1} )  +
 E^{8,3}\otimes d (E^{5,2}+E^{6,1} )\\[3mm]
&&c_0^5= E^{1,2}\otimes d (E^{4,3}-E^{7,8} )
  +E^{6,5}\otimes d (E^{4,3}-E^{7,8} )+
   E^{1,4}\otimes d (E^{2,3}-E^{7,6} )\\
&&
  +E^{8,5}\otimes d (E^{2,3}-E^{7,6} ) +
   E^{3,4}\otimes d (E^{2,1}-E^{5,6} )
  +E^{8,7}\otimes d (E^{2,1}-E^{5,6} ) \\
&&+
   E^{3,2}\otimes d (E^{4,1}-E^{5,8} )
  +E^{6,7}\otimes d (E^{4,1}-E^{5,8} ) +
  E^{1,7}\otimes d (E^{2,8}-E^{4,6} )\\
&&
 +E^{3,5}\otimes d (E^{2,8}-E^{4,6} ) +
  E^{6,4}\otimes d(E^{5,3}+E^{7,1} )  +
   E^{8,2}\otimes d (E^{5,3}+E^{7,1} )\\[3mm]
&&c_0^6= E^{1,2}\otimes d (E^{3,4}-E^{8,7} )
 +E^{6,5}\otimes d (E^{3,4}-E^{8,7} ) +
  E^{1,3}\otimes d (E^{2,4}-E^{8,6} )\\
&&
  +E^{7,5}\otimes d (E^{2,4}-E^{8,6} ) +
  E^{4,2}\otimes d (E^{3,1}-E^{5,7} )
  +E^{6,8}\otimes d (E^{3,1}-E^{5,7} ) \\
&&+
   E^{4,3}\otimes d (E^{2,1}-E^{5,6} )
  +E^{7,8}\otimes d (E^{2,1}-E^{5,6} ) +
  E^{1,8}\otimes d (E^{2,7}-E^{3,6} )\\
&&
  +E^{4,5}\otimes d (E^{2,7}-E^{3,6} ) +
  E^{6,3}\otimes d (E^{5,4}+E^{8,1} ) +E^{7,2}\otimes d (E^{5,4}+E^{8,1} )\\[3mm]

\text{$\deg=1$}&:& \text{as multiplication by the matrix }
 d_i=\begin{pmatrix}0&0\\E^{i,i}&0\end{pmatrix} \text{ for any $n>2$ }  (i=1,\ldots,n)\\[2mm]
&&\text{For $n=2$, there are also four cocycles given by}\\
&&c_1^1= E^{1,2}\otimes d (E^{3,8}-E^{4,7} )
  +E^{6,5}\otimes d (E^{3,8}-E^{4,7} ) +
   E^{1,3}\otimes d (E^{2,8}-E^{4,6} )\\
&&
  +E^{7,5}\otimes d (E^{2,8}-E^{4,6} ) +
  E^{1,4}\otimes d (E^{2,7}-E^{3,6} )
   +E^{8,5}\otimes d (E^{2,7}-E^{3,6} ) \\
&&+
 E^{7,4}\otimes d (E^{2,1}-E^{5,6} )  +
 E^{8,3}\otimes d (E^{2,1}-E^{5,6} )  +
 E^{6,3}\otimes d (E^{4,1}-E^{5,8} )  \\
&&+
  E^{7,2}\otimes d (E^{4,1}-E^{5,8} ) +
  E^{6,4}\otimes d (E^{3,1}-E^{5,7} )  +
   E^{8,2}\otimes d (E^{3,1}-E^{5,7} )\\[3mm]
&&c_1^2= E^{2,3}\otimes d (E^{1,8}-E^{4,5} )
  +E^{7,6}\otimes d (E^{1,8}-E^{4,5} ) +
  E^{2,4}\otimes d (E^{1,7}-E^{3,5} ) \\
&&+E^{8,6}\otimes d (E^{1,7}-E^{3,5} ) +
  E^{2,1}\otimes d (E^{3,8}-E^{4,7} )
+E^{5,6}\otimes d (E^{3,8}-E^{4,7} )\\
&& +
 E^{7,4}\otimes d (E^{1,2}-E^{6,5} )  +
  E^{8,3}\otimes d (E^{1,2}-E^{6,5} )  +
   E^{5,3}\otimes d (E^{4,2}-E^{6,8} ) \\
&& +
    E^{7,1}\otimes d (E^{4,2}-E^{6,8} )  +
    E^{5,4}\otimes d (E^{3,2}-E^{6,7} )  +
    E^{8,1}\otimes d (E^{3,2}-E^{6,7}   )\\[3mm]
&&c_1^3= E^{3,4}\otimes d (E^{1,6}-E^{2,5} )
+E^{8,7}\otimes d (E^{1,6}-E^{2,5} ) +
  E^{3,1}\otimes d (E^{2,8}-E^{4,6} ) \\
&&+E^{5,7}\otimes d (E^{2,8}-E^{4,6} ) +
   E^{3,2}\otimes d (E^{1,8}-E^{4,5} )
+E^{6,7}\otimes d (E^{1,8}-E^{4,5} ) \\
&&+
  E^{6,4}\otimes d (E^{1,3}-E^{7,5} ) +
   E^{8,2}\otimes d (E^{1,3}-E^{7,5} )  +
   E^{5,2}\otimes d (E^{4,3}-E^{7,8} )  \\
&&+
   E^{6,1}\otimes d (E^{4,3}-E^{7,8} )  +
   E^{5,4}\otimes d (E^{2,3}-E^{7,6} )  +
   E^{8,1}\otimes d (E^{2,3}-E^{7,6}  )\\[3mm]
&&c_1^4=  E^{4,1}\otimes d (E^{2,7}-E^{3,6} ) +E^{5,8}\otimes d (E^{2,7}-E^{3,6} ) +
   E^{4,2}\otimes d (E^{1,7}-E^{3,5} )\\
&&+E^{6,8}\otimes d (E^{1,7}-E^{3,5} ) +
  E^{4,3}\otimes d (E^{1,6}-E^{2,5} )
+E^{7,8}\otimes d (E^{1,6}-E^{2,5} )\\
&& +
 E^{6,3}\otimes d (E^{1,4}-E^{8,5} ) +
   E^{7,2}\otimes d (E^{1,4}-E^{8,5} )  +
 E^{5,2}\otimes d (E^{3,4}-E^{8,7} ) \\
&& +
   E^{6,1}\otimes d (E^{3,4}-E^{8,7} ) +
  E^{5,3}\otimes d (E^{2,4}-E^{8,6} )  +
   E^{7,1}\otimes d (E^{2,4}-E^{8,6} )  \\[3mm]

\text{$\deg=2$}&:&
   E^{7,4} \otimes d (E^{1,6}-E^{2,5} )
+E^{8,3} \otimes (d(E^{1,6}-E^{2,5} )
+E^{6,3} \otimes d (E^{1,8}-E^{4,5}  )\\
&&+E^{7,2} \otimes d (E^{1,8}-E^{4,5} )+
E^{6,4} \otimes d(E^{1,7}-E^{3,5} )+
E^{8,2} \otimes d (E^{1,7}-E^{3,5} )\\
&&+E^{5,2}\otimes  d (E^{3,8}-E^{4,7} )+
E^{6,1} \otimes  d (E^{3,8}-E^{4,7} )
+E^{5,3} \otimes d (E^{2,8}-E^{4,6} )\\
&&+E^{7,1} \otimes d (E^{2,8}-E^{4,6} )
+E^{5,4} \otimes  d(E^{2,7}-E^{3,6} )
+E^{8,1} \otimes d (E^{2,7}-E^{3,6})
\end{array}
\end{equation}
}

\end{Lemma}

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\end{thebibliography}


\label{lastpage}

\end{document}

\DL{just one comment: nothing "is clear" to me}:

1) trifles:  we index cocycles by weight (with respect to the outer
derivatives), and numerate the cocycle of the same weight (better
say "degree") with a superscript.

2) more serious. In \S7 of \cite{LeP} it is written what should we
take for $\fpe(n)$ if $p=2$. Then come its subalgebras $\fspe(n)$
(is it simple if $n$ is odd $\geq 3$?), and simple subquotient
$\fspe(2n)/\fc$ (is it simple if $n\geq 2$?)

2a) why didn't you consider n odd?!

3) Did I misunderstand you? I thought that you were going to insert
the results of $H^2(\fg)$ (with coeff. in the trivial module) in the
cases not covered by eq. (3), see a few lines I've just added there.

\DL{I inserted several subsections from \S7 mentioned above.}

1307.1858.extracted_bib
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