on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of is given. 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 . Minimal ideals Since W is isotropic, multiplication of elements of W inside is skew. Hence vectors in W anti-commute, and is just the exterior algebra Λ∗W. Consequently, the k-fold product of W with itself, Wk, is one-dimensional. Let ω be a generator of Wk. In terms of a basis of in W, one possibility is to set Note that (i.e., ω is nilpotent of order 2), and moreover, for all . The following facts can be proven easily: If , then the left ideal is a minimal left ideal. Furthermore, this splits into the two spin spaces and on restriction to the action of the even Clifford algebra. If , then the action of the unit vector u on the left ideal decomposes the space into a pair of isomorphic irreducible eigenspaces (both denoted by Δ), corresponding to the respective eigenvalues +1 and −1. In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in . We shall show that I must be equal to by proving that it contains a nonzero scalar multiple of ω. Fix a basis wi of W and a complementary basis wi′ of W so that Note that any element of I must have the form αω, by virtue of our assumption that . Let be any such element. Using the chosen basis, we may write where the ai1...ip are scalars, and the Bj are auxiliary elements of the Clifford algebra. Observe now that the product Pick any nonzero monomial a in the