On triple homomorphisms of Lie algebras

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