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