This theorem states that (V,(cdot,cdot)) be a finite-dimensional real Euclidean [vector space](/page/Vector%20Space), let subset V be a finite reduced crystallographic root system, and let = 1,dots, n be an ordered simple system for .. 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.