We show in a certain Lie-algebra, the connections between the Lie subalgebra G＋ ：= G-t-G＊ 4- [G, G＊], generated by a Lie subalgebra G, and the properties of G. This allows us to investigate some useful information about the structure of such two Lie subalgehras. Some results on the relations between the two Lie subalgebras are obtained. As an application, we get the following conclusion： Let ,4 C /3（2d） be a space of self-adjoint operators and L： ：= A ~ i^4 the corresponding complex Lie＊-algebra. G＋ = G 4- G＊ 4- [G, G＊] and G are two LM-decomposable Lie subalgebras of L： with the decomposition 6＋ = 7^（6＋） 4- S, G -- T~~ 4- 86, and T^6 C T^（6＋）. Then 6＋ is ideally finite iff T~ ：= 7~6 4- ~ 4- [T^6,7~] is a quasisolvable Lie subalgebra, S^- ：= 86 4- S~ 4- [$6, $~] is an ideally finite semisimple Lie subalgebra, and [7~6,86] = [R.~, 86] = {0}.
