Computing split maximal toral subalgebras of Lie algebras over fields of small characteristic

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 (Ryba, 2007; Cohen and Murray, 2009), 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 (Cohen and Roozemond, 2009) these algorithms are very useful for the recognition of reductive Lie algebras over such fields.