English

Abstract

Let $$\mathcal{N}$$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra L over an arbitrary field $$\mathbb{F}$$ , there exists a smallest ideal I of L such that L/I ∈ $$\mathcal{N}$$ . This uniquely determined ideal of L is called the nilpotent residual of L and is denoted by L $$\mathcal{N}$$ . In this paper, we define the subalgebra S(L) = ∩H≤LIL(H $$\mathcal{N}$$ ). Set S0(L) = 0. Define Si+1(L)/Si(L) = S(L/Si(L)) for i > 1. By S∞(L) denote the terminal term of the ascending series. It is proved that L = S∞(L) if and only if L $$\mathcal{N}$$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra L with S(L) = L.