Cln(C) is central simple and so by the Artin–Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes not only the scalars but the pseudoscalars (degree n elements) as well. We can always find a normalized pseudoscalar ω such that . Define the operators These two operators form a complete set of orthogonal idempotents, and since they are central they give a decomposition of Cln(C) into a direct sum of two algebras where The algebras are just the positive and negative eigenspaces of ω and the P± are just the projection operators. Since ω is odd, these algebras are mixed by α (the linear map on V defined by ): and therefore isomorphic (since α is an automorphism). These two isomorphic algebras are each central simple and so, again, isomorphic to a matrix algebra over C. The sizes of the matrices can be determined from the fact that the dimension of Cln(C) is 2n. What we have then is the following table: The even subalgebra of Cln(C) is (non-canonically) isomorphic to Cln−1(C). When n is even, the even subalgebra can be identified with the block diagonal matrices (when partitioned into 2×2 block matrix). When n is odd, the even subalgebra are those elements of for which the two factors are identical. Picking either piece then gives an isomorphism with . Real case 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. Classification of quadratic forms Firstly, there are non-isomorphic quadratic forms of a given degree, classified by signature. Every nondegenerate quadratic form on a real vector space is equivalent to the standard diagonal form: where is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,q. The Clifford algebra on Rp,q is denoted Clp,q(R). A standard orthonormal basis {ei} for Rp,q consists of mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1. Unit pseudoscalar The unit pseudoscalar in Clp,q(R) is defined as This is both a Coxeter element of sorts (product of reflections) and a longest element of a Coxeter group in the Bruhat order; this is an analogy. It corresponds to and generalizes a volume form (in the