Triple and generalized triple derivations of the parabolic subalgebras of the general linear Lie algebra over a commutative ring

Let R be an arbitrary commutative ring with identity and be the general linear Lie algebra over R. Let P be a parabolic subalgebra of . The main aim of this article is to describe triple derivations and generalized triple derivations of P. We show that any triple derivation of P can be uniquely expressed as a sum of standard triple derivations and any generalized triple derivation of P can be expressed as a sum of a scalar map and a triple derivation.