A NOTE ON DERIVATIONS OF LIE ALGEBRAS
Abstract 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 \[ L^n\subseteq \sum _{d\in D}d(L), \] for some n>1, is necessarily solvable. As a result, we show that if L has a derivation d:L→L such that Ln⊆d(L), for some n>1, then L is solvable.