Simple vectorial Lie algebras in characteristic 2 and their superizations

The list of simple finite dimensional Lie algebras over an algebraically closed field of characteristic 2 is wider than that in other characteristics. In particular, it contains desuperizations of modular analogs (whose description is often far from being obvious) of complex simple vectorial Lie superalgebras both exceptional and serial. We describe 15 Weisfeiler gradings of the 5 exceptional families and one W-grading for each of 2 series (with 2 exceptional subseries) of analogs in characteristic 2 of simple complex Lie algebras. For each of these Lie algebras obtained as desuperization of a Lie superalgebra, and for the Shen Lie algebra (see arXiv:1307.1551), we establish the number of parameters the unconstrained shearing vector depends on. Two of the 15 exceptions are completely new. One of the exceptional Lie algebras is a deform of the Lie algebra of divergence-free vector fields; the deformed algebra is not isomorphic to the initial one as might happen in positive characteristic even for deforms given by cocycle of nontrivial cohomology class. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. The simple Lie superalgebras, that can be obtained by these two methods from simple Lie algebras we describe here, are new.