On the Nilpotency of a Pair of Lie Algebras

Abstract

‎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.