Explicit isomorphisms of real Clifford algebras

It is well known that the Clifford algebra Cl p,q associated to a nondegenerate quadratic form on R n (n = p + q) is isomorphic to a matrix algebra K(m) or direct sum K(m) ⊕ K(m) of matrix algebras, where K = R, C, H. On the other hand, there are no explicit expressions for these isomorphisms in literature. In this work, we give a method for the explicit construction of these isomorphisms. Let F be a field and let V be a finite-dimensional vector space over F and Q : V → F a quadratic form on V. The Clifford algebra Cl(V ,Q) is an associative algebra with unit 1, which contains and is generated by V , with v · v = Q(v) · 1 for all v ∈ V. Formally, one can define the Clifford algebra Cl(V ,Q) as follows. Definition 1.1. The Clifford algebra Cl(V ,Q) associated to a vector space V over F with quadratic form Q can be defined as Cl(V ,Q) = T(V) I(Q) , (1.1) where T(V) is the tensor algebra T(V) = F ⊕ V ⊕ (V ⊗ V) ⊕ ··· and I(Q) is the two-sided ideal in T(V) generated by elements v ⊗ v − Q(v) · 1. Just like the tensor algebra and the exterior algebra, the Clifford algebra has the following universal property. Theorem 1.2. Given an associative unital F-algebra A (with unit 1) and a linear map f : V → A with f (v) · f (v) = Q(v) · 1 for all v ∈ V , then there is a unique homomorphism