On triple homomorphisms of Lie algebras

Let L and K be two Lie algebras over a commutative ring with identity. In this paper, under some conditions on L and K, it is proved that every triple homomorphism from L onto K is the sum of a homomorphism and an antihomomorphism from L
into K. We also show that a finite-dimensional Lie algebra L over an algebraically closed field of characteristic zero is nilpotent of class at most 2 if and only if the sum of every homomorphism and every antihomomorphism on L is a triple homomorphism.