On the equality of triple derivations and derivations of Lie algebras

Abstract

Let $L$ be a Lie algebra over a commutative ring with identity. In the present paper, under some mild conditions on $L$, it is proved that every triple derivation of $L$ is a derivation. In particular, we show that in perfect Lie algebras and free Lie algebras every triple derivation is a derivation. Finally we apply our results to show that every triple derivation of the Lie algebra of block upper triangular matrices is a derivation.