Finite gradings of Lie algebras

We show that the algebra Der(L) of derivations of a strongly nondegenerate Lie algebra L graded by an ordered group G with a finite grading (and satisfying a mild technical condition) inherits the grading from L, i.e., Der(L), which turns out to be a strongly nondegenerate Lie algebra, is G-graded and its support has the same length as that of L. This result follows by considering Der(L) as a subalgebra of the graded maximal algebra of quotients of the Lie algebra L. We specialize the result when L is a Lie algebra of the form A−/ZA or K/ZK, for A a semiprime associative algebra, K the Lie algebra of the skew elements of a semiprime associative algebra with involution, and ZA and ZK their respective centers.