This theorem states that g be a finite-dimensional semisimple Lie algebra over an algebraically closed field k of characteristic zero, let h subset g be a Cartan subalgebra, and let := \ h \ 0\ : g 0\ be the set of non-zero roots for the root space decomposition of g relative to h.. 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.