that has 1 on the -th spot and zero elsewhere. Hence, each root is of the form and the root space decomposition is the decomposition of matrices: Noncompact case Suppose is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a Cartan involution and let be the eigenspace decomposition of , where are the eigenspaces for 1 and -1, respectively. For example, if and the negative transpose, then . Let be a maximal abelian subspace. Now, consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the restricted root space decomposition): where the elements in are called the restricted roots, for any linear functional ; in particular, , . Moreover, is a root system but not necessarily reduced one (i.e., it can happen are both roots). The case of sl(n,C) If , then may be taken to be the diagonal subalgebra of , consisting of diagonal matrices whose diagonal entries sum to zero. Since has dimension , we see that has rank . The root vectors in this case may be taken to be the matrices with , where is the matrix with a 1 in the spot and zeros elsewhere. If is a diagonal matrix with diagonal entries , then we have . Thus, the roots for are the linear functionals given by . After identifying with its dual, the roots become the vectors in the space of -tuples that sum to zero. This is the root system known as in the conventional labeling. The reflection associated to the root acts on by transposing the and diagonal entries. The Weyl group is then just the permutation group on elements, acting by permuting the diagonal entries of matrices in . Generalizations Semisimple Lie algebras admit certain generalizations. Firstly, many statements that are true for semisimple Lie algebras are true more generally for reductive Lie algebras. Abstractly, a reductive Lie algebra is one whose adjoint representation is completely reducible, while concretely, a reductive Lie algebra is a direct sum of a semisimple Lie algebra and an abelian Lie algebra; for example, is semisimple, and is reductive. Many properties of semisimple Lie algebras depend only on reducibility. Many properties of complex semisimple/reductive Lie algebras