A Construction of Gradings of Lie Algebras

In this article we present a method to construct gradings of Lie algebras. It requires the existence of an abelian inner ideal B of the Lie algebra whose subquotient, a Jordan pair, is covered by a finite grid, and it produces a grading of the Lie algebra L by the weight lattice of the root system associated to the covering grid. As a corollary one obtains a finite Z-grading L = L-n circle plus ... circle plus L-n such that B = L-n. In particular, our assumption on B holds for abelian inner ideals of finite length in nondegenerate Lie algebras.