---
abstract: 'This article classifies the real forms of Lie Superalgebra by Vogan diagrams, developing Borel de Seibenthal theorem of semisimple Lie algebras for Lie superalgebras. A Vogan diagram is a Dynkin diagram of triplet $(\mathfrak{g}_{C},\mathfrak{h_{\overline{0}}},\triangle^{+})$, where $\mathfrak{g}_{C}$ is a real Lie superalgebra, $\mathfrak{h_{\overline{0}}}$ cartan subalgebra, $\triangle^{+}$ positive root system. Although the classification of real forms of contragradient Lie superalgebras already furnished but this article’s method is a quicker one to classify.'
author:
- 'B Ransingh$^{*}$ and K C Pati'
title: A quick proof of classification of real forms of Basic Lie superalgebras by Vogan diagrams 
---

Department of Mathematics\
National Institute of Technology\
Rourkela (India)

Email$^{*}$- bransingh@gmail.com

2010 AMS Subject Classification : 17B05, 17B22, 17B40\
Keywords : Lie superalgebras, Vogan diagrams

Introduction
============

For a complex semi-simple Lie algebra $\mathfrak{g}$, it is well known that the conjugacy classes of real forms of $\mathfrak{g}$ are in one to one correspondence with the conjugacy classes of involutions of $\mathfrak{g}$, if we associate to a real form $\mathfrak{g}_{\mathbb{R}}$ one of its Cartan involutions $\theta$. Using a suited pair $(\mathfrak{h}, \prod)$ of a Cartan subalgebra $\mathfrak{h}$ and a basis $\prod$ of the associated root system, an involution is described by a “Vogan diagram”.

Knapp [@Knapp:Lie; @groups] brought Vogan diagrams of simple Lie algebras into the light to represent the real forms of the complex simple Lie algebras. Batra [@batra:affine; @batra:vogan] developed a corresponding theory of Vogan diagrams for almost compact real forms of indecomposable nontwisted affine Kac-Moody Lie algebras. Tanushree [@tanushree:twisted] developed the theory of Vogan diagrams for almost compact real forms of indecomposable twisted affine Kac-Moody Lie algebras. Similar theory is also developed to find out the Vogan diagrams of hyperbolic Kac-Moody algebras[@ransingh:pati].

A Vogan diagram is a Dynkin diagram with some additional information as follows the 2-elements orbits under $\theta$ (Cartan involution) are exhibited by joining the corresponding simple roots by a double arrow and the 1-element orbit is painted in black (respectively, not painted), if the corresponding imaginary simple root is noncompact (respectively compact).

The real form is defined as a real Lie superalgebra such that its complexification is the original complex Lie superalgebra. It can be seen easily that every standard real form is naturally associated to an antilinear involutive automorphism of the complex Lie superalgebra.

The classification of real semisimple Lie algebras will use maximally compact and split Cartan subalgebras, The Vogan diagram is based on the classification of maximally compact Cartan subalgebras.

Recently similar work has been done using Vogan superdiagrams to classify the real forms of contragradient Lie superalgebras[@chuah:vsuper], where the extended Dynkin diagrams of Lie superalgebra is used. But our method uses the ordinary Dynkin diagram as done by Knapp[@Knapp:Lie; @groups]. In this article we construct all the real forms of Lie superalgebras by Vogan diagrams.

Preliminary
===========

The general and special linear Lie superalgebras.
-------------------------------------------------

Let $V=V_{\overline{0}}\oplus V_{\overline{1}}$ be a vector superspace, so that End$(V)$ is an associative superalgebra. The End$(V)$ with the supercummutator forms a Lie superalgebra, called the general linear Lie superalgebra and is denoted by $\mathfrak{gl}(m|n)$, where $V=\mathbb{C}^{m|n}$. With respect to an suitable ordered basis End$(V)$ and $\mathfrak{gl}(m|n)$ can be realized as $(m+n)\times(m+n)$ complex matrices of the block form.

$\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)$ where $a$ , $b$, $c$ and $d$ are respectivily $m\times m$, $m\times n$, $n\times m$ and $n\times n$ matrices. The even subalgebra of $\mathfrak{gl}(m|n)$ is $\mathfrak{gl}(m)\oplus\mathfrak{gl}(n)$ , which consists of matrices of the form $\left(\begin{array}{cc}
a & 0\\
0 & d
\end{array}\right)$, While the odd subspace consists of $\left(\begin{array}{cc}
0 & b\\
c & 0
\end{array}\right)$

A Lie superalgebras $\mathfrak{g}$ is an algebra graded over $\mathbb{Z}_{2}$ , i.e., $\mathfrak{g}$ is a direct sum of vector spaces $\mathfrak{g}=\mathfrak{g}_{\overline{0}}\oplus \mathfrak{g}_{\overline{1}}$, and such that the bracket satisfies

1.  $[\mathfrak{g}_{i},
    \mathfrak{g}_{j}]\subset\mathfrak{g} _{i+j(mod2)}$,

2.  $[x,y]=-(-1)^{|x||y|}[y,x]$, (Skew supersymmetry) $\forall$ homogenous $x,y,z\in \mathfrak{g}$ (Super Jacobi identity)

3.  $[x,[y,z]]=[[x,y],z]+\left(-1\right)^{|x||y|}[y,[x,z]]\forall z\in \mathfrak{g}$

A bilinear form $(.,.):\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{C}$ on a Lie superalgebra is called **invariant** if $([x,y],z)=(x,[y,z])$, for all $x,y,z\in \mathfrak{g}$

The Lie superalgebra $\mathfrak{g}$ has a root space decomposition with respect to $\mathfrak{h}$ $$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\triangle}\mathfrak{g}_\alpha$$

A root $\alpha$ is even if $\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{\overline{0}}$ and it is odd if $\mathfrak{g}_{\alpha}\subset\mathfrak{g}_{\overline{1}}$

A *Cartan subalgebra* $\mathfrak{h}$ of diagonal matrices of $\mathfrak{g}$ is defined to be a Cartan subalgebra of the even subalgebra $\mathfrak{g}_{\overline{0}}$. Since every inner automorphism of $\mathfrak{g}_{\overline{0}}$ extends to one of Lie superalgebra $\mathfrak{g}$ and Cartan subalgebras of $\mathfrak{g}_{\overline{0}}$ are conjugate under inner automorphisms. So the Cartan subalgebras of $\mathfrak{g}$ are conjugate under inner automorphism.

 Real forms of Basic Lie superalgebras
--------------------------------------

\[[@Parker:classification] Proposition 1.4\] Let $\mathcal{\mathfrak{g}}$ be a complex classical Lie superalgebra and let $C$ be an involutive semimorphism of $\mathcal{\mathfrak{g}}$. Then $\mathcal{\mathfrak{g}}_{C}=\left\{ x+Cx|x\in\mathfrak{g}\right\} $ is a real classical Lie superalgebra.

\[[@Parker:classification] Proposition 1.5\] If $\mathcal{\mathfrak{g}_C}$ is a real classical Lie superalgebra, its complexification $\mathfrak{g}=\mathfrak{g}_{C}\otimes\mathbb{C}$ is a Lie superalgebra which is either classical or direct sum of two isomorphic ideals which are classical

\[[@Parker:classification] Theorem 4\] Up to isomorphism, the real forms of the classical Lie superalgebras are uniquely determined by the real form $\mathcal{\mathfrak{g}}_{\overline{0}C}$ of the Lie subalgebra $\mathcal{\mathfrak{g}}_{\overline{0}}$.

The real form is said to standard (graded) when the real structure is standard (graded). Let $\mathfrak{g}_{C}$ be a real form of $\mathfrak{g}$ and let $\omega$ be the corresponding complex conjugation. Then $\omega|_{\mathfrak{g}_{\overline{0}}}$ is an antilinear involution of the Lie algebra $\mathfrak{g}_{\overline{0}}$. Hence there is a corresponding Cartan decomposition $\mathfrak{g}_{\overline{0}}=t_{\overline{0}}\oplus p_{\overline{0}}$, with Cartan involution $C$.

The following table gives a list of all the real forms associated with basic classical Lie superalgebras.

  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
    $\mathfrak{g}$    $\mathfrak{g}_{\overline{0}}$                                      $\mathfrak{g}_C$                           $\mathfrak{g}_{\overline{0}C}$
  ------------------- ------------------------------------------------------------------ ------------------------------------------ ---------------------------------------------------------------------------------------------------
       $A(m,n)$       $\mathfrak{sl}(m)\oplus \mathfrak{sl}(n)\oplus U(1)$               $\begin{array}{l}                          $\begin{array}{l}
                                                                                         \mathfrak{sl}(m|n;\mathbb{R})\\            \mathfrak{sl}(m,\mathbb{R})\oplus \mathfrak{sl}(n,\mathbb{R})\oplus\mathbb{R}\\
                                                                                         \mathfrak{sl}(m|n;\mathbb{H})\\            \mathfrak{su}^{*}(m)\oplus \mathfrak{su}^{*}(n)\oplus\mathbb{R}\\
                                                                                         \mathfrak{su}(p,m-p|q,n-q)                 \mathfrak{su}(p,m-p)\oplus \mathfrak{su}(q,n-q)\oplus i\mathbb{R}
                                                                                         \end{array}$                               \end{array}$

       $A(n,n)$       $\mathfrak{sl}(n)\oplus \mathfrak{sl}(n)$                          $\begin{array}{l}                          $\begin{array}{l}
                                                                                         \mathfrak{psl}(n|n;\mathbb{R})\\           \mathfrak{sl}(n,\mathbb{R})\oplus \mathfrak{sl}(n,\mathbb{R})\\
                                                                                         \mathfrak{psl}(n|n;\mathbb{H})\\           \mathfrak{su}^{*}(n)\oplus \mathfrak{su}^{*}(n)\\
                                                                                         \mathfrak{su}(p.n-p|q,n-q)                 \mathfrak{su}(p,n-p)\oplus \mathfrak{su}(q,n-q)
                                                                                         \end{array}$                               \end{array}$

   $\begin{array}{l}  $\begin{array}{l}                                                  $\begin{array}{l}                          $\begin{array}{l}
       B(m,n)\\       \mathfrak{so}(2m+1)\oplus \mathfrak{sp}(2n)\\                      \mathfrak{osp}(p,2m+1-p|2n;\mathbb{R})\\   \mathfrak{so}(p,2m+1-p)\oplus \mathfrak{sp}(2n;\mathbb{R})\\
        B(0,n)        \mathfrak{sp}(2n)                                                  \mathfrak{osp}(1|2n;\mathbb{R})            \mathfrak{sp}(2n,\mathbb{R})
     \end{array}$     \end{array}$                                                       \end{array}$                               \end{array}$

       $C(n+1)$       $\mathfrak{so}(2)\oplus \mathfrak{sp}(2n)$                         $\begin{array}{l}                          $\begin{array}{l}
                                                                                         \mathfrak{osp}(p|2n;\mathbb{R})\\          \mathfrak{so}^{*}(2)\oplus \mathfrak{sp}(2n;\mathbb{R})\\
                                                                                         \mathfrak{osp}(2|2q,2n-2q;\mathbb{H})      \mathfrak{so}^{*}(2)\oplus \mathfrak{sp}(2q,2n-2q)
                                                                                         \end{array}$                               \end{array}$

       $D(m,n)$       $\mathfrak{so}(2m)\oplus \mathfrak{sp}(2n)$                        $\begin{array}{l}                          $\begin{array}{l}
                                                                                         \mathfrak{osp}(p,2m-p|2n;\mathbb{R})\\     \mathfrak{so}(p,2m-p)\oplus \mathfrak{sp}(2n;\mathbb{R})\\
                                                                                         \mathfrak{osp}(2m|2q,2n-2q;\mathbb{H})     \mathfrak{so}^{*}(2m)\oplus \mathfrak{sp}(2q,2n-2q)\\
                                                                                         \end{array}$                               \mathfrak{sp}(n)\oplus \mathfrak{so}^{*}(2m)\\
                                                                                                                                    \mathfrak{so}(2m)\oplus \mathfrak{so}^{*}(2n,\mathbb{R})
                                                                                                                                    \end{array}$

        $F(4)$        $\mathfrak{sl}(2)\oplus \mathfrak{so}(7)$                          $\begin{array}{l}                          $\begin{array}{l}
                                                                                         F(4;0)\\                                   \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)\\
                                                                                         F(4;3)\\                                   \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(1,6)\\
                                                                                         F(4;2)\\                                   \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(2,5)\\
                                                                                         F(4;1)                                     \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(3,4)
                                                                                         \end{array}$                               \end{array}$

        $G(3)$        $\mathfrak{sl}(2)\oplus G_{2}$                                     $\begin{array}{l}                          $\begin{array}{l}
                                                                                         G(3,0)\\                                   \mathfrak{sl}(2,\mathbb{R})\oplus G_{2,0}\\
                                                                                         G(3,1)                                     \mathfrak{sl}(2,\mathbb{R})\oplus G_{2,2}
                                                                                         \end{array}$                               \end{array}$

    $D(2,1;\alpha)$   $\mathfrak{sl}(2)\oplus \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)$   $\begin{array}{l}                          $\begin{array}{l}
                                                                                         D(2,1;\alpha;0)\\                          \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{sl}(2,\mathbb{R})\\
                                                                                         D(2,1;\alpha;1)\\                          \mathfrak{sl}(u)\oplus \mathfrak{sl}(u)\oplus \mathfrak{sl}(u)\\
                                                                                         D(2,1;\alpha;2)                            \mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{R})
                                                                                         \end{array}$                               \end{array}$
  -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Vogan diagrams of Basic Lie Superalgebras
=========================================

The following Lemma shows the existence of

\[Lemma 4.7 [@kac:denominator]\]There exists $C\in Aut(\mathfrak{g})$ such that $C|_{g_{0}}$ is a nontrivial automorphism if and only if $\mathfrak{g}$ is of type $A(m,n)$, $D(m,n)$, $D(2,1;\alpha)$ ,$\alpha\in\{1,(-2)^{\pm 1}\}$ and $C|_{g_{0}}$is as follows.

1.  If $\mathfrak{g}$ is of type $A(m,n)$ with $n\ne m$, then $C|_{\mathfrak{g}_0}$ restricts to the nontrivial diagram automorphism of both $A_n$ and $A_m$.

2.  If $\mathfrak{g}$ is of type $A(n,n)$, then $C|_{\mathfrak{g}_{0}}$ is either the nontrivial diagram automorphism of both $A_n$ components, or it is the flip automorphism between the two $A_n$ components, or the composition of these two automorphisms.

3.  If $\mathfrak{g}$ is of type $D(m,n),\,m>2$, then $C|_{\mathfrak{g}_0}$ is the unique diagram automorphism of $\mathfrak{g}_0$.

4.  If $\mathfrak{g}$ is of type $D(2,1)\cong D(2,1,a)$, $a\in{1, (-2)^{\pm1} }$, then $C$ is the unique diagram automorphism of the diagram.

The Vogan diagram of Lie superalgebras is the Vogan diagram of the even part of Lie superalgerbas. In addition to that

-   The vertices fixed by the Cartan involution of the even part is painted (or unpainted) depending whether the the root is noncompact (or compact).

-   Label the 2- elements orbit by the diagram automorphism indicated with two sided arrow.

-   The odd root remain unchanged.

An abstract Vogan diagram is an abstract Dynkin diagram with two pieces of additional structure , one is an automorphism of order 1 or 2 of the diagram, which is to be indicated by labeling the 2 element orbits. The other is the subset of the 1 element orbits which is to be indicated by painting the vertices corresponding to the members of the subset of noncompact roots. Every Vogan diagram is of course an abstract Vogan diagram of Lie superalgebra.

If an abstract Vogan diagram is given, then there exist a real Lie superalgebra $\mathfrak{g}_{C}$, a Cartan involution $\theta$, a maximally compact $\theta$ stable Cartan subalgebra and a positive system $\triangle_{\overline{0}}^{+}$ for $\triangle=\triangle(\mathfrak{g,h})$ that takes $it_{\overline{0}}$ before $a_{\overline{0}}$ such that the given diagram is the Vogan diagram of $(\mathfrak{g}_{C},\mathfrak{h}_{\overline{0}},\triangle_{0}^{+})$.

Briefly the theorem says that any abstract Vogan diagram comes from some $\mathcal{\mathfrak{g}}_{\overline{0}}$. Thus the theorem is an analog for real semisimple Lie algebras of the existence theorem for complex semisimple Lie algebras.

We will modify the Borel and de Siebenthal Theorem for Lie superalgebra.

Let $\mathfrak{g}_{C}$ be a non complex real Lie superalgebra and Let the Vogan diagram of $\mathfrak{g}_{C}$ be given that correponding to the triple $(\mathfrak{g}_{C},\mathfrak{h}_{0},\triangle^{+})$. Then $\exists$ a simple system $\prod'$ for $\triangle=\triangle(\mathcal{\mathfrak{g}},\mathfrak{h})$, with corresponding positive system $\triangle^{+}$, such that $(\mathcal{\mathfrak{g}}_
{C},\mathfrak{h}_{\overline{0}},\triangle^{+})$ is a triple and there is at most two painted simple root in its Vogan diagrams of $A(m,n),D(m,n)$ and at most three painted vertices in $D(2,1;\alpha)$. Furthermore suppose the automorphism associated with the Vogan diagram is the identity,that $\prod'={\alpha_{1},\cdots,\alpha_{l}}$ and that ${\omega_{1},\cdots,\omega_{l}}$ is the dual basis for each even part such that $\left\langle\omega_{j},\alpha_{k}\right\rangle=\delta_{jk}/\epsilon_{kk} $, where $\epsilon_{kk}$ is the diagonal entries to make cartan matrix symmetric. The the double painted simple root of even parts may be chosen so that there is no $i'$ with $\left\langle \omega_{i}-\omega_{i'},\omega_{i'}\right\rangle>0$ for each even part.

We know $\mathfrak{g}=\mathcal{\mathfrak{g}}_{\bar{0}}\oplus\mathcal{\mathfrak{g}}_{\bar{1}}$. The positive even root system $\triangle_{0}^{+}$ can be written as $$\triangle_{0}^{+}=\triangle_{01}^{+}\cup\triangle_{02}^{+}$$ where $\triangle_{01}^{+}$ are the even positive root system for simple root system formed by $e_{i}$ basis and $\triangle_{02}^{+}$ are for $\delta_{j}$ basis. For the even part, we take $<\omega_{i},\alpha_{j}>=\delta_{ij}/\epsilon_{kk}$ . This makes the Cartan matrix symmetric and so that we can get the inverse of cartan matrix of $A_{m}$ and $A_n$ for $A(m,n)$ Lie superalgebra. Similarly construction will follows for other Lie superalgebras. Each inverse for even part is associted with the dual basis $\omega$. For the odd part the condition is $<\omega_{i},\alpha_{j}>=\delta_{ij}$ and donot get any painted vertices. The Symmetrizable condition of Kac-Cartan matrix gives $S=\epsilon_{kk} A$, where $S$ is the symmetric cartan matrix. The below table gives the values of $\epsilon_{kk}$ for different superalgebras.

   Lie superalgebra             $\epsilon_{kk}$
  ------------------ -------------------------------------
       $A(m,n)$           $(1,\cdots,1,-1,\cdots,-1)$
       $B(m,n)$          $(1,\cdots,1,-1\cdots,-1,-2)$
       $B(0,n)$                 $1,\cdots,1,2$
        $C(n)$           $(-1,1,\cdots,1,\frac{1}{2})$
       $D(m,n)$       $(1,\cdots,1,-1-1,\cdots,-1,-1,-1)$
   $D(2,1;\alpha)$           $(1,-1,\frac{1}{a})$
        $F(4)$              $ (-1,1,\frac{1}{2}) $
        $G(3)$          $(-\frac{1}{2},1,\frac{1}{3},)$

Taking suitable normalization condition for each type of Lie superalgebras and from the two Lemmas 6.97 and 6.98 [@Knapp:Lie; @groups] we get redudancy test for each even part. So now the Vogan diagram of Lie superalgebras becomes two painted vertices Vogan diagram.

Let $\bigtriangleup$ be an irreducible abstract reduced root system in a real vector superspace $V$, let $\Pi_{01}$ and $\Pi_{02}$ be the two simple simple root system for even parts $e_{i}$ and $\delta_{j}$ basis respectivily and let $\omega$ and $\omega'$ be nonzero members of $V$ that are domiant relative to $\Pi_{i}'$s. Then $\left\langle \omega,\omega'\right\rangle >0$.

Using the suitable normalisations of $e_i$ and $\delta_j$ we get the proof of the Lemma.

Let $\mathcal{\mathfrak{g}}_{0C}$ be a noncomplex simple real Lie superalgebra and let the Vogan diagram of $\mathcal{\mathfrak{g}}_{0C}$ be given that corresponding to the triple $(\mathfrak{g}_{0},\mathfrak{h}_{0},\triangle^{+})$. Write $\mathfrak{h}_{01}=\mathfrak{t}_{01}\oplus\mathfrak{a}_{01}$ and $\mathfrak{h}_{02}=\mathfrak{t}_{02}\oplus\mathfrak{a}_{02}$ for two even parts. . Let $V$ be the span of simple roots that are imaginary, let $\triangle_{0}$ be the root system $\triangle\cap V$, let $\mathcal{H}$ be the subset of $it_{0}$ paired with $V$ and let $\Lambda$ be the subset of $\mathcal{H}$ where all roots of $\triangle_{0}$ take integer values and all noncompact roots of $\triangle_{0}$ take odd integer values. Then $\Lambda$ is nonempty. In fact if $\alpha_{1},\cdots,\alpha_{m}$ is any simple system for $\triangle_{0}$ and if $\omega_{1},\cdots,\omega_{m}$ in $V$ are defined by $\left\langle \omega,\alpha_{k}\right\rangle =\delta_{jk}$, then the element

$$\omega=\underset{i\mbox{ with }\alpha_{i}\mbox{ noncompact}}{\sum}\omega_{i}$$

Fix a simple system $\alpha_{1},\cdots,\alpha_{m}$ for $\bigtriangleup_{\overline{0}}$ and let $\bigtriangleup^{+}_{\overline{0}}$ be the set of positive roots of $\bigtriangleup_{\overline{0}}$. Define $\omega_{1},\cdots,\omega_{m}$ by $\left\langle \omega,\alpha_{k}\right\rangle =\delta_{jk}$. If $\alpha=\sum_{i=1}^{m} 
 n_{i}\alpha_{i}$ is a positive root of $\bigtriangleup_{\overline{0}}$, then $\left\langle \omega,\alpha\right\rangle $ is the sum of the $n_{i}$ for which $\alpha_{i}$ is noncompact.

Using induction of the Lemma 6.98 [@Knapp:Lie; @groups] for even part of root system the above Lemma will be proved and each even roots satisfy

compact root + compact root = compact root

compact root+ noncompact root = noncompact root

noncompact root + noncompact root = noncompact root

1.  $A(m,n)$

    The Vogan diagrams and real foms of Lie superalgebras $A(m,n)$ are as follows.

    $$\begin{picture}(60,10) \thicklines 
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     \put(-15,-25){\makebox(0,0){$\mathfrak{sl}(n,\mathbb{C})$}}
       \end{picture}$$

    $$\begin{picture}(60,10) \thicklines 
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     \put(-65,-25){\makebox(0,0){$\mathfrak{su}(p,m-p)$}}\put(65,-25){\makebox(0,0){$\mathfrak{su}(r,n-r)$}}
       \end{picture}$$

    $$\begin{picture}(60,10) \thicklines 
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        \put(-75,-25){\makebox(0,0){$\mathfrak{sl}(m,\mathbb{R})$}}\put(55,-25){\makebox(0,0){$\mathfrak{sl}(n,\mathbb{R})$}}
       \end{picture}$$

    $$\begin{picture}(60,10) \thicklines 
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       \qbezier(39,-6)(66,-16)(93, -6)
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     \put(-95,-28){\makebox(0,0){$\mathfrak{su}^{*}(m)$}}\put(65,-28){\makebox(0,0){$\mathfrak{su}^{*}(n)$}}
       \end{picture}$$

2.  $B(m,n)$ The Lie subalgebra of $\mathfrak{g}_{0}$ is $C_{m}\oplus B_{n}$ The only trivial automophism of even part of Vogan diagram of $B(m,n)$ is shown below and the real form is $\mathfrak{sp}(2n,\mathbb{R})\oplus \mathfrak{so}(p,2m+1-p)$

    $$\begin{picture}(60,10) \thicklines 
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     \put(39,0){\circle{9}}\put(66,0){\circle{9}}
     \put(-92,0){\line(1,0){19}}\put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)}
     \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}}
    \put(45,-25){\makebox(0,0){$so(p,2m+1-p)$}}
       \end{picture}$$

    Because of missing of real form of the first even part, we need and additional $C_{n}$ Dynkin diagram superimposed Vogan diagrams below. $$\begin{picture}(60,10) \thicklines 
     \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle*{9}} 
     \put(39,0){\circle{9}}\put(66,0){\circle{9}}
    \put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)}
     \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}}
     \put(-65,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}}\put(45,-25){\makebox(0,0){$so(p,2m+1-p)$}}
     \put(-100,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

    The another real form of Lie superalgebra $B(m,n)$ is $sp(2n,\mathbb{R})\oplus so(2m+1)$ and the corresponding Vogan diagram is drawn below. $$\begin{picture}(60,10) \thicklines 
     \put(-97,0){\circle{9}}\put(-70,0){\circle{9}} \put(-43,0){\circle{9}} \put(-20,-2.5){$\bigotimes$}\put(12,0){\circle{9}} 
     \put(39,0){\circle{9}}\put(66,0){\circle{9}}
    \put(-65,0){\dottedline{4}(1,0)(17,0)}\put(-38,0){\line(1,0){19}}\put(-11,0){\dottedline{4}(1,0)(18,0)}
     \put(16,0){\dottedline{4}(1,0)(18,0)}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}}
     \put(-65,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}}\put(45,-25){\makebox(0,0){$so(2m+1)$}}
     \put(-100,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

3.  Case $B(0,n)$

    The Vogan diagram below is a unpainted diagram but it consists of its own painted vertices on the extreme right.\
    $$\begin{picture}(60,20) \thicklines 
    \put(-43,0){\circle{9}} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle*{9}}
     \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\line(1,0){19}}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}}
      \put(25,-25){\makebox(0,0){$sp(2n,\mathbb{R})$}}
       \end{picture}$$

4.  Case $C(n+1)$

    The unpainted Vogan diagram of $C(n+1)$ creats the real form $so^{*}(2)\oplus sp(2n,\mathbb{R})$\
    $$\begin{picture}(60,20) \thicklines 
    \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}}
     \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\dottedline{4}(1,0)(19,0)}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}
       \end{picture}$$

    The trivial automophism of the even part of $C(n+1)$ makes the Vogan diagram below and the real form is $so^{*}(2)\oplus sp(2q,2n-2q)$\
    $$\begin{picture}(60,20) \thicklines 
    \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle*{9}} \put(39,0){\circle{9}}\put(66,0){\circle{9}}
     \put(-38,0){\line(1,0){19}}\put(-12,0){\dottedline{4}(1,0)(19,0)}\put(16,0){\dottedline{4}(1,0)(19,0)}\put(37,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}
       \end{picture}$$

5.  Case $D(m,n)$. The Lie subalgebra of $\mathfrak{g}_{0}$ is $C_{m}\oplus D_{n}$ . The compact real form of $C_{m}$ is $\mathfrak{sp}(m)$. The real form $sp(2n,\mathbb{R})\oplus so(2m)$ of the abstract Vogan diagram of $D(m,n)$ for the above subalgebras is followed below.

    $$\begin{picture}(-20,20)\thicklines
      \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} 
      \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
     \put(-132,0){\dottedline{4}(1,0)(19,0)} \put(-105,0){\dottedline{4}(1,0)(19,0)}
     \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}
     \put(10,-2){\line(3,-2){16}}
    \put(5,-25){\makebox(0,0){$\mathfrak{so}(2m)$}}
         \end{picture}$$

    Since from the diagram we get only $\mathfrak{so}(2m)$ part of real form, so for the $\mathfrak{sp}(2n)$ part we need the addition $C_{n}$ diagram in the diagram above. Subsequently the Vogan diagram becomes

    $$\begin{picture}(-20,20)\thicklines
      \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} 
      \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
    \put(-105,0){\dottedline{4}(1,0)(19,0)}
     \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)}
     \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
       \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2n,\mathbb{R})$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}(2m)$}}
       \put(-139,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

    The first trivial involution for one of the even part for the Vogan diagram of $D(m,n)$ is given below and the real form of this diagram is $\mathfrak{sp}(2n,\mathbb{R})\oplus \mathfrak{so}(p,2m-p)$.

    $$\begin{picture}(-20,20)\thicklines
      \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle*{9}} 
      \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
      \put(-105,0){\dottedline{4}(1,0)(19,0)}
     \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)}
     \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
      \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2n,\mathbb{R})$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}(p,2m-p)$}}
       \put(-139,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

    The nontrivial involution for the Vogan diagram of $D(m,n)$ is given below and the real form of this diagram is $sp(2q,2n-2q)\oplus so^{*}(2m)$.

    $$\begin{picture}(-20,20)\thicklines
       \put(-165,0){\circle{9}}
     \put(-137,0){\circle{9}} \put(-137,0){\circle{9}}\put(-109,0){\circle*{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} 
      \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
     \put(-132,0){\dottedline{4}(1,0)(19,0)} \put(-105,0){\dottedline{4}(1,0)(19,0)}
     \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
       \qbezier(35,16)(51,1)(35, -16)
       \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}}
       \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(2q,2n-2q)$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}^{*}(2m)$}}
          \put(-167,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

    The below Vogan diagram is formed by the nontrivial involution . The real form for this superlagebra is $\mathfrak{sp}(n)\oplus \mathfrak{so}^{*}(2m)$.

    $$\begin{picture}(-20,20)\thicklines
      \put(-137,0){\circle{9}}\put(-109,0){\circle{9}}\put(-81,0){\circle{9}}\put(-56,-2.5){$\bigotimes$} \put(-22,0){\circle{9}} 
      \put(5,0){\circle{9}}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
     \put(-105,0){\dottedline{4}(1,0)(19,0)}
     \put(-76,0){\line(1,0){20}} \put(-46,0){\dottedline{4}(1,0)(19,0)}\put(-18,0){\dottedline{4}(1,0)(19,0)} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
       \qbezier(35,16)(51,1)(35, -16)
       \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}}
       \put(-105,-25){\makebox(0,0){$\mathfrak{sp}(n)$}}\put(5,-25){\makebox(0,0){$\mathfrak{so}^{*}(2m)$}}
       \put(-139,0){\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(20,0){\line(-1,1){10}}\put(20,0){\line(-1,-1){10}}
      \end{picture}}\end{picture}$$

6.  Case $D(2,1;\alpha)$

    The unpainted and no two element orbit Vogan diagram is given below with the real form. Since we can get only the real form $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ from ordinary Vogan diagram $$\begin{picture}(40,20)\thicklines
    \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
        \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
           \put(100,0){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{su}(2)$}}
      \end{picture}$$

    So our requisite Vogan diagrams for the suitable real forms are $$\begin{picture}(40,20)\thicklines
     \put(-22,0){\circle{9}} \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
       \put(-18,0){\line(1,0){19}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
           \put(100,0){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{su}(2)\oplus \mathfrak{su}(2)$}}
      \end{picture}$$ $$\begin{picture}(40,20)\thicklines
     \put(-22,0){\circle*{9}}  \put(0,-2.5){$\bigotimes$}\put(30,16){\circle*{9}} \put(30,-16){\circle*{9}} 
       \put(-18,0){\line(1,0){19}}  \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
                  \put(105,0){\makebox(0,0){$\mathfrak{sl}(2,\mathbb {R})\oplus\mathfrak{ sl}(2,\mathbb {R})\oplus
                  \mathfrak{sl}(2,\mathbb {R})$}}
      \end{picture}$$

    The nontrivial involution of the Dynkin diagram of $D(2,1;\alpha)$ makes the following Vogan diagram as shown below.

    $$\begin{picture}(40,20)\thicklines
     \put(-22,0){\circle*{9}}  \put(0,-2.5){$\bigotimes$}\put(30,16){\circle{9}} \put(30,-16){\circle{9}} 
     \put(-18,0){\line(1,0){19}} \put(10,2){\line(3,2){16}}\put(10,-2){\line(3,-2){16}}
       \qbezier(35,16)(51,1)(35, -16)
       \put(35,16){\vector( -1, 1){0}}\put(35,-16){\vector( -1, -1){0}}
       \put(120,0){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{C})\oplus \mathfrak{sl}(2,\mathbb{R})$}}
      \end{picture}$$

7.  Case F(4) From the Dynkin diagram we can get only the real form $so(7)$ and the Vogan diagram

    $$\begin{picture}(60,20) \thicklines 
    \put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}
     \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}\put(16,0){\line(1,0){19}}
     \put(0,-32){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)$}} 
       \end{picture}$$

    we add the extra even part root to get the desired real forms and Vogan diagrams. $$\begin{picture}(60,20) \thicklines 
    \put(-75,0){\circle*{9}}\put(-48,-3){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}
    \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}\put(16,0){\line(1,0){19}}
     \put(0,-32){\makebox(0,0){$\mathfrak{sl}(2,\mathbb{R})\oplus \mathfrak{so}(7)$}} 
       \end{picture}$$ $$\begin{picture}(60,20) \thicklines 
    \put(-75,0){\circle{9}}\put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle*{9}}\put(12,0){\circle{9}} \put(39,0){\circle{9}}
     \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}\put(16,0){\line(1,0){19}}
     \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(1,6)$}} 
       \end{picture}$$ $$\begin{picture}(60,20) \thicklines 
    \put(-75,0){\circle{9}}\put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle*{9}} \put(39,0){\circle{9}}
      \put(-71,0){\line(1,0){24}}\put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}\put(16,0){\line(1,0){19}}
      \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(2,5)$}} 
       \end{picture}$$ $$\begin{picture}(60,20) \thicklines 
    \put(-75,0){\circle{9}} \put(-48,-2.5){$\bigotimes$} \put(-15,0){\circle{9}}\put(12,0){\circle{9}} \put(39,0){\circle*{9}}
      \put(-71,0){\line(1,0){24}} \put(-38,0){\line(1,0){19}}\put(-18,0){{\begin{picture}(20,20)
      \put(6,-2){\line(1,0){20}}\put(6,2){\line(1,0){20}}
      \put(15,0){\line(1,-1){10}}\put(15,0){\line(1,1){10}}
      \end{picture}}}\put(16,0){\line(1,0){19}}
      \put(0,-32){\makebox(0,0){$\mathfrak{su}(2)\oplus \mathfrak{so}(3,4)$}} 
       \end{picture}$$

8.  Case $G(3)$

    The Vogan diagram of $G(3)$ with real form $\mathfrak{sl}(2,\mathbb{R})\oplus G_{2,0}$ and $\mathfrak{sl}(2,\mathbb{R})\oplus G_{2,2}$ are

    $$\begin{picture}(60,20) \thicklines
    \put(-34,0){\circle*{9}}\put(-7,-2.5){$\bigotimes$}\put(26,0){\circle{9}}\put(54,0){\circle{9}}
    \put(-30,0){\line(1,0){24}}\put(3,0){\line(1,0){19}}\put(23,0){{\begin{picture}(20,20)
      \put(7,0){\line(1,0){20}}
      \put(6,-3){\line(1,0){22}}\put(6,3){\line(1,0){22}}
      \put(15,0){\line(1,1){10}}\put(15,0){\line(1,-1){10}}
      \end{picture}}}
    \end{picture}$$

    $$\begin{picture}(60,20) \thicklines
    \put(-34,0){\circle*{9}}\put(-7,-2.5){$\bigotimes$}\put(26,0){\circle{9}}\put(54,0){\circle*{9}}
    \put(-30,0){\line(1,0){24}}\put(3,0){\line(1,0){19}}\put(23,0){{\begin{picture}(20,20)
      \put(7,0){\line(1,0){20}}
      \put(6,-3){\line(1,0){22}}\put(6,3){\line(1,0){22}}
      \put(15,0){\line(1,1){10}}\put(15,0){\line(1,-1){10}}
      \end{picture}}}
    \end{picture}$$

#### Acknowledgement:

The authors thank National Board of Higher Mathematics, India (Project Grant No. 48/3/2008-R&DII/196-R) for financial support.

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