This theorem states that L be a nonzero finite-dimensional semisimple Lie algebra over an algebraically closed field k of characteristic zero, let H subset L be a Cartan subalgebra, and let = \ H \ 0\ : L 0\ be the root system of L relative to H, where L = \ x L : [h,x] = (h)x h H\ .. It is used in the structure and classification of finite-dimensional Lie algebras, especially in arguments involving Cartan data, root systems, Weyl groups, and Dynkin diagrams.