Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber , we mean the convex cone where is a unique vector such that . The criterion then reads: if and only if, for each positive root , (1) is an integer and (2) lies in . A linear functional satisfying the above equivalent condition is called a dominant integral weight. Hence, in summary, there exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple -modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character formula. The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra is completely reducible; i.e., it is a direct sum of simple -modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra. Real semisimple Lie algebra For a semisimple Lie algebra over a field that has characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic zero. But over the field of real numbers, there are still the structure results. Let be a finite-dimensional real semisimple Lie algebra and the complexification of it (which is again semisimple). The real Lie algebra is called a real form of . A real form is called a compact form if the Killing form on it is negative-definite; it is necessarily the Lie algebra of a compact Lie group (hence, the name). Compact case Suppose is a compact form and a maximal abelian subspace. One can show (for example, from the fact is the Lie algebra of a compact Lie group) that consists of skew-Hermitian matrices, diagonalizable over with imaginary eigenvalues. Hence, is a Cartan subalgebra of and there results in the root space decomposition (cf. #Structure) where each is real-valued on ; thus, can be identified with a real-linear functional on the real vector space . For example, let and take the subspace of all diagonal matrices. Note . Let be the linear functional on given by for . Then for each , where is the matrix