  We describe some recent computer investigations with the `Constraint Logic
Programming over Finite Domains' -- CLP(FD) -- library in the Prolog
programming environment to search for new simple Lie algebras over the field
$\GF(2)$ of $2$ elements. Motivated by a paper of Grishkov et. al., we
specifically look for those with a `thin decomposition', and we settle one of
their conjectures. We extrapolate from our results the existence of two new
infinite families of simple Lie algebras, in addition to finding seven new
sporadic examples in dimension $31$. We also better contextualise some
previously discovered simple algebras, putting them into families which do not
seem to have ever appeared in the literature, and give an updated table of
those currently known.
