This theorem states that (W,S) be a Coxeter system with simple reflections S=s i : i in I, let = + sqcup (- +) be its associated real root system with simple roots = i : i in I, and let ell:Wto Ncup0 denote the Coxeter length function with respect to S.. 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.