for each , is diagonalizable. As it turns out, is abelian and so all the operators in are simultaneously diagonalizable. For each linear functional of , let . (Note that is the centralizer of .) Then (The most difficult item to show is . The standard proofs all use some facts in the representation theory of ; e.g., Serre uses the fact that an -module with a primitive element of negative weight is infinite-dimensional, contradicting .) Let with the commutation relations ; i.e., the correspond to the standard basis of . The linear functionals in are called the roots of relative to . The roots span (since if , then is the zero operator; i.e., is in the center, which is zero.) Moreover, from the representation theory of , one deduces the following symmetry and integral properties of : for each , Note that has the properties (1) and (2) the fixed-point set is , which means that is the reflection with respect to the hyperplane corresponding to . The above then says that is a root system. It follows from the general theory of a root system that contains a basis of such that each root is a linear combination of with integer coefficients of the same sign; the roots are called simple roots. Let , etc. Then the elements (called Chevalley generators) generate as a Lie algebra. Moreover, they satisfy the relations (called Serre relations): with , . The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the is a Cartan matrix). This is a theorem of Serre. In particular, two semisimple Lie algebras are isomorphic if they have the same root system. The implication of the axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence, "all possible" semisimple Lie algebras (finite-dimensional over an algebraically closed field of characteristic zero). The Weyl group is the group of linear transformations of generated by the 's. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of are invariant under the Weyl group. Example root space decomposition in sln(C) For and the Cartan subalgebra of diagonal matrices, define by , where denotes the diagonal matrix