exterior algebra; for the trivial quadratic form, the unit pseudoscalar is a volume form), and lifts reflection through the origin (meaning that the image of the unit pseudoscalar is reflection through the origin, in the orthogonal group). To compute the square , one can either reverse the order of the second group, yielding , or apply a perfect shuffle, yielding . These both have sign , which is 4-periodic (proof), and combined with , this shows that the square of ω is given by Note that, unlike the complex case, it is not in general possible to find a pseudoscalar that squares to +1. Center If n (equivalently, ) 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, ) 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 (equivalently, if ) 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 (equivalently, if ) 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. Classification All told there are three properties which determine the class of the algebra Clp,q(R): signature mod 2: n is even/odd: central simple or not signature mod 4: : if not central simple, center is or C signature mod 8: the Brauer class of the algebra (n even) or even subalgebra (n odd) is R or H Each of these properties depends only on the signature modulo 8. The complete classification table is given below. The size of the matrices is determined by the requirement that Clp,q(R) have dimension 2p+q. It may be seen that of all matrix ring types mentioned, there is only one type shared between both complex and real algebras: the type M(2m,C). For example, Cl2(C) and Cl3,0(R) are both determined to be M2(C). It is important to note that there is a difference in the classifying isomorphisms used. Since the Cl2(C) is algebra isomorphic via a C-linear map (which is necessarily R-linear),