Lie Algebras Generated by Extremal Elements

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type An (n ≥ 1), Bn (n ≥ 3), Cn (n ≥ 2), Dn (n ≥ 4), En (n = 6, 7, 8), F4 and G2 are shown to be n + 1, n + 1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.