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 fieldk 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.Research supported in part by Hungarian National Foundation for Scientific Research grants T016503 and F4116