This theorem states that be a finite irreducible root system in a real Euclidean [vector space](/page/Vector%20Space) E with inner product (cdot,cdot), let subset be a base of simple roots, and let + be the corresponding set of positive roots.. 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.