A Lie algebra variation on a theorem of Wedderburn

Abstract

THEOREM. Let A be a finite dimensional associative algebra over a field F. Suppose that A has a basis over F consisting of nilpotent elements. Then A itself must be nilpotent. Given a finite dimensional Lie algebra L which is a Lie subalgebra of an associative algebra A (possibly infinite dimensional), we can ask what results from supposing that L has a basis over F consisting of elements that are nilpotent in the associative algebra. We cannot conclude that L is nilpotent and finite dimensional. In fact, this is rarely the case. For example,