On permutably complemented subalgebras of finite dimensional Lie algebras

Let L be a finite-dimensional Lie algebra. We say a subalgebra H of L is permutably complemented in L if there is a subalgebra K of L such that L=H+K and H∩K=0. Also, if every subalgebra of L is permutably complemented in L, then L is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in particular, we obtain some characterizations for supersolvability of a finite-dimensional Lie algebra in terms of permutably complemented subalgebras.