On the nilpotent residuals of all subalgebras of Lie algebras

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≤LI_L(H(mathcal{N})). Set S₀(L) = 0. Define S_i+1(L)/S_i(L) = S(L/S_i(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.