Computing Cartan subalgebras of lie algebras

We consider the algorithmic problem of computing Cartan subalgebras in Lie algebras over finite fields and algebraic number fields. We present a deterministic polynomial time algorithm for the case when the ground field k is sufficiently large. Our method is based on a solution of a linear algebra problem: the task of finding a locally regular element in a subspace of linear transformations. Also, we give a polynomial time algorithm for restricted Lie algebras over arbitrary finite fields. Both methods require an auxiliary procedure for finding non-nilpotent elements in subalgebras. This problem is also treated. Computational experiences are discussed at the end of the paper.