On Lie algebras of classical type

Abstract

G. B. Seligman has proved in [1] the following result: If V is a simple restricted Lie algebra over an algebraically closed field of characteristic p> 7, and if 2 has a restricted representation with nondegenerate trace form, then V is of classical type. By an algebra of classical type is meant an analogue over a field of characteristic p of one of the simple Lie algebras (including the five exceptional algebras) of characteristic 0; for the precise statement, see [1]. We shall show here that the above result of Seligman may be proved without the assumption of restrictedness of the algebra and its representation. We begin with a lemma on matrices. Let Sn(hW) denote the space of all nXn matrices over a field !, and consider this as a Lie algebra (under commutation) over .