  Important subalgebras of a Lie algebra of an algebraic group are its toral
subalgebras, or equivalently (over fields of characteristic 0) its Cartan
subalgebras. Of great importance among these are ones that are split: their
action on the Lie algebra splits completely over the field of definition. While
algorithms to compute split maximal toral subalgebras exist and have been
implemented [Ryb07, CM09], these algorithms fail when the Lie algebra is
defined over a field of characteristic 2 or 3. We present heuristic algorithms
that, given a reductive Lie algebra L over a finite field of characteristic 2
or 3, find a split maximal toral subalgebra of L. Together with earlier work
[CR09] these algorithms are very useful for the recognition of reductive Lie
algebras over such fields.
