This theorem states that V be a finite-dimensional real [inner product space](/page/Inner%20Product%20Space) with inner product (cdot,cdot), let subset V be a crystallographic root system, and let = 1,dots, n be a simple root basis of .. 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.