As above, we let (V, g) be an n -dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification V ⊗ ⊗ R C and let g denote the induced bilinear form on V ⊗ ⊗ R C . Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g | W = 0. If n = 2 k is even, then let W ′ be an isotropic subspace complementary to W. If n = 2 k + 1 is odd, let W ′ be a maximal isotropic subspace with W ∩ W ′ = 0, and let U be the orthogonal complement of W ⊕ W ′. In both the even- and odd-dimensional cases W and W ′ have dimension k. In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u.