As above, we let be an -dimensional complex vector space equipped with a nondegenerate bilinear form. If is a real vector space, then we replace by its complexification and let denote the induced bilinear form on . Let be a maximal isotropic subspace, i.e. a maximal subspace of such that . If is even, then let be an isotropic subspace complementary to . If is odd, let be a maximal isotropic subspace with , and let be the orthogonal complement of . In both the even- and odd-dimensional cases and have dimension . In the odd-dimensional case, is one-dimensional, spanned by a unit vector .
As above, we let be an -dimensional complex vector space equipped with a nondegenerate bilinear form. If is a real vector space, then we replace by its complexification and let denote the induced bilinear form on . Let be a maximal isotropic subspace i.e. a maximal subspace of such that . If is even, then let be an isotropic subspace complementary to  If is odd, let be a maximal isotropic subspace with , and let be the orthogonal complement of . In both the even- and odd-dimensional cases and have dimension . In the odd-dimensional case, is one-dimensional, spanned by a unit vector 