On the nilpotency of a pair of Lie algebras

In this paper, we introduce the new concepts of nilpotency, upper and lower central series for a pair of Lie algebras (L, N), in which N is an ideal of Lie algebra L. We study the properties of these concepts and prove the analogue of Robinson Theorem for a pair of Lie algebras plays an important role to find the results connecting to the idea of nilpotency in the pair of Lie algebras. In particular we find a criterion such that an extension of pair of Lie algebras can be nilpotent.