A NOTE ON DERIVATIONS OF LIE ALGEBRAS

In this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D <= Der(L), with the property L(n) subset of Sigma(d epsilon D)d(L), for some n > 1, is necessarily solvable. As a result, we show that if L has a derivation d: L -> L such that L(n) subset of d(L), for some n > 1, then L is solvable.