Lie Algebras Represented as a Sum of Two Subalgebras

Let L be a Lie algebra represented as a sum of two subalgebras A and B. We prove that if L belongs to a subclass of the class of locally finite Lie algebras over a field of characteristic ≠ 2 and both A and B are locally nilpotent, then L is locally soluble. We also prove that if L is a serially finite Lie algebra over a field of characteristic zero, then any common serial subalgebra of A and B is serial in L.