This theorem states that g be a finite-dimensional semisimple Lie algebra over an algebraically closed field k of characteristic 0, let h subset g be a Cartan subalgebra, let subset h be the corresponding root system, and let := \ x : [h,x] = (h)x h \ denote the root space for in .. 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.