Extension of a result by Schur for Lie algebras

Let L be a Lie algebra and ID(L) be the set of all derivations of L whose images lie in the drived algebra of L. In this paper, the central ideal S (L) = fx 2 L j (x) = 0; 8 2 ID(L)g of a Lie algebra L is studied. We prove some results analogous to a result by Schur and a weak form of its converse in the context of ID-derivations. Also, we give an upper bound for dimension of L S (L) in some special conditions.