  Over algebraically closed fields of positive characteristic, for simple Lie
(super)algebras, and certain Lie (super)algebras close to simple ones, with
symmetric root systems (such that for each root, there is minus it of the same
multiplicity) and of ranks less than or equal to 8 - most needed in an approach
to the classification of simple vectorial Lie superalgebras (i.e., Lie
superalgebras realized by means of vector fields on a supermanifold), - we list
the outer derivations and nontrivial central extensions. When the conjectural
answer is clear for the infinite series, it is given for any rank. We also list
the outer derivations and nontrivial central extensions of one series of
non-symmetric (except when considered in characteristic 2), namely periplectic,
Lie superalgebras - the one that preserves the nondegenerate symmetric odd
bilinear form, and of the Lie algebras obtained from them by desuperization. We
also list the outer derivations and nontrivial central extensions of an analog
of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several
results indigenous to positive characteristic are of particular interest being
unlike known theorems for characteristic 0, some results are, moreover,
counterintuitive.
