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

Computing Split Maximal Toral Subalgebras of Lie algebras over Fields of Small Characteristic