  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).
