Classification of Clifford algebras

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In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified as rings. 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 Cl_1,1(R) and Cl_2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.