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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 Cl1,1(R) and Cl2,0(R), which are both isomorphic as rings to the ring of two by two matrices over the real numbers.

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, and other structure, such as the distinguished subspace of generators V, are not used here. This article uses the (+) sign convention for Clifford multiplication so that

for all vectors v in the vector space of generators V, where Q is the quadratic form on the vector space V. We will denote the algebra of n × n matrices with entries in the division algebra K by Mn(K) or End(Kn). The direct sum of two such identical algebras will be denoted by 1=Mn(K) ⊕ Mn(K), which is isomorphic to Mn(K ⊕ K).

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.

The classification allows Dirac spinors and Weyl spinors to be defined in even dimension. In even dimension n, the Clifford algebra Cln(C) is isomorphic to End(CN), which has its fundamental representation on 1=Δn := CN. A complex Dirac spinor is an element of Δn. The term complex signifies that it is the element of a representation space of a complex Clifford algebra, rather than that is an element of a complex vector space. The even subalgebra Cln0(C) is isomorphic to End(CN/2) ⊕ End(CN/2) and therefore decomposes to the direct sum of two irreducible representation spaces , each isomorphic to CN/2. A left handed (respectively right handed) complex Weyl spinor is an element of Δ (respectively, Δ).

The structure theorem is simple to prove inductively. For base cases, Cl0(C) is simply C ≅ End(C), while Cl1(C) is given by the algebra C ⊕ C ≅ End(C) ⊕ End(C) by defining the only gamma matrix as 1=γ1 = (1, −1). We will also need Cl2(C) ≅ End(C2). The Pauli matrices can be used to generate the Clifford algebra by setting 1=γ1 = σ1, 1=γ2 = σ2. The span of the generated algebra is End(C2). The proof is completed by constructing an isomorphism Cln+2(C) ≅ Cln(C) ⊗ Cl2(C). Let γa generate Cln(C), and generate Cl2(C). Let 1=ω = i be the chirality element satisfying 1=ω2 = 1 and 1=ω + ω = 0. These can be used to construct gamma matrices for Cln+2(C) by setting 1=Γa = γa ⊗ ω for 1 ≤ a ≤ n and 1=Γa = 1 ⊗ for 1=a = n + 1, n + 2. These can be shown to satisfy the required Clifford algebra and by the universal property of Clifford algebras, there is an isomorphism Cln(C) ⊗ Cl2(C) → Cln+2(C). Finally, in the even case this means by the induction hypothesis Cln+2(C) ≅ End(CN) ⊗ End(C2) ≅ End(CN+1). The odd case follows similarly as the tensor product distributes over direct sums.

The real case is significantly more complicated, exhibiting a periodicity of 8 rather than 2, and there is a 2 parameter family of Clifford algebras.

If n (equivalently, 1=p − q) is even, the algebra Clp,q(R) is central simple and so isomorphic to a matrix algebra over R or H by the Artin–Wedderburn theorem. If n (equivalently, 1=p − q) is odd then the algebra is no longer central simple but rather has a center which includes the pseudoscalars as well as the scalars. If n is odd and 1=ω2 = +1 (equivalently, if 1=p − q ≡ 1 (mod 4)) then, just as in the complex case, the algebra Clp,q(R) decomposes into a direct sum of isomorphic algebras

each of which is central simple and so isomorphic to matrix algebra over R or H.

If n is odd and 1=ω2 = −1 (equivalently, if 1=p − q ≡ −1 (mod 4)) then the center of Clp,q(R) is isomorphic to C and can be considered as a complex algebra. As a complex algebra, it is central simple and so isomorphic to a matrix algebra over C.