Computing subalgebras and $\mathbb{Z}_2$-gradings of simple Lie algebras over finite fields

This paper introduces two new algorithms for Lie algebras over finite fields and applies them to the investigate known simple of dimension at most $20$ field $\mathbb{F}_2$ with elements. The first algorithm is a approach towards construction $\mathbb{Z}_2$-gradings algebra characteristic $2$. Using this, we observe that each has $\mathbb{Z}_2$-grading determine associated superalgebras. second allows us compute all subalgebras field. We apply this subalgebras, maximal subquotients $16$ (with exception $15$-dimensional Zassenhaus algebra).