Let formula_1 be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of formula_1 can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition, a Cartan subalgebra (also called a maximal toral subalgebra) formula_37 of formula_1 is a maximal subalgebra such that, for each formula_39, formula_40 is diagonalizable. As it turns out, formula_37 is abelian and so all the operators in formula_42 are simultaneously diagonalizable. For each linear functional formula_43 of formula_37, let