This theorem states that be a reduced crystallographic root system in a real Euclidean [vector space](/page/Vector%20Space), let = 1,dots, r be a choice of simple roots, and let W be the Weyl group generated by the simple reflections s i := s i for 1 leq i leq r.. 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.