Proof. Let be the linear map with . It commutes with the action of and satisfies , so , where are subrepresentations. The quotient maps and of Definitions 9.10.3 and 9.10.4 restrict to isomorphisms and : the first map kills exactly the span of the tensors , which is , and the second kills exactly the span of the tensors , which is , because .
Step 1: . Fix and a basis of consisting of eigenvectors of (Lemma 13.1.5), with eigenvalues . Then has the basis () and has the basis (), and each of these vectors is an eigenvector of with eigenvalue . Hence Averaging over and using (Proposition 13.5.1) gives Step 1.
Step 2: invariant forms. A bilinear form on is the same as a linear functional on , via (Theorem 9.8.6). The form is -invariant if and only if for all and , that is, if and only if is a -invariant element of the dual representation . The form is symmetric if and only if , that is, vanishes on ; and is skew-symmetric if and only if vanishes on . Since , restriction to identifies the symmetric forms with , and restriction to identifies the skew-symmetric forms with , compatibly with the actions of . Since , and likewise . So , where is the dimension of the space of -invariant symmetric bilinear forms on and that of the space of -invariant skew-symmetric ones.
Step 3. By Theorem 16.5.4: if is not real-valued, and . If is real, the invariant forms are the multiples of one symmetric form, so , , and . If is quaternionic, they are the multiples of one skew-symmetric form, so , , and . ∎