operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then where the signs are given by the following table: Clifford scalar product When the characteristic is not 2, the quadratic form Q on V can be extended to a quadratic form on all of (which we also denoted by Q). A basis-independent definition of one such extension is where ⟨a⟩ denotes the scalar part of a (the degree-0 part in the Z-grading). One can show that where the vi are elements of V – this identity is not true for arbitrary elements of . The associated symmetric bilinear form on is given by One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of is nondegenerate if and only if it is nondegenerate on V. The operator of left (respectively right) Clifford multiplication by the transpose a of an element a is the adjoint of left (respectively right) Clifford multiplication by a with respect to this inner product. That is, and Structure of Clifford algebras In this section we assume that characteristic is not 2, the vector space V is finite-dimensional and that the associated symmetric bilinear form of Q is nondegenerate. A central simple algebra over is a matrix algebra over a (finite-dimensional) division algebra with center . For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. If V has even dimension then is a central simple algebra over K. If V has even dimension then the even subalgebra is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. If V has odd dimension then is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. If V has odd dimension then the even subalgebra is a central simple algebra over K. The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that has even dimension and a non-singular bilinear form with discriminant , and suppose that is another vector space with a quadratic form. The Clifford algebra of is