On Gradings Modulo $2$ of Simple Lie Algebras in Characteristic $2$

The ground field in the text is of characteristic $2$. The classification of modulo $2$ gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson–Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than $3$ indeterminates. Here we completely describe gradings modulo $2$ for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo $2$ gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).