In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl2,0(R) and Cl1,1(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers. Notation and conventions The Clifford product is the manifest ring product for the Clifford algebra, and all algebra homomorphisms in this article are with respect to this ring product. Other products defined within Clifford algebras, such as the exterior product, are not used here. This article uses the (+) sign convention for Clifford multiplication so that for all vectors , where Q is the quadratic form on the vector space V. We will denote the algebra of matrices with entries in the division algebra K by Mn(K) or . The direct sum of two such identical algebras will be denoted by , which is isomorphic to . Bott periodicity Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group. Complex case The complex case is particularly simple: every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form where , so there is essentially only one Clifford algebra in each dimension. This is because the complex numbers include by which and so positive or negative terms are equivalent. We will denote the Clifford algebra on Cn with the standard quadratic form by Cln(C). There are two separate cases to consider, according to whether n is even or odd. When n is even the algebra