The Clifford algebra and the Chevalley map - a computational approach ( summary

One of the basic properties of the Clifford algebra gives an explicit basis for it in terms of a basis of the underlying vector space (Theorem 1 below), and another one provides a vector space isomorphism between the Clifford algebra and the exterior algebra of the same vector space (the so-called Chevalley map, Theorem 2 below). While both of these properties appear in standard literature such as [1] and [2], sadly I have never seen them proven in the generality they deserve (they hold over arbitrary commutative rings rather than just fields of characteristic 0, at least as long as we are talking about bilinear rather than quadratic forms). Besides, some proofs found in literature are sloppily written or otherwise unsatisfactory. Here we are going to present a computational proof of both of these properties, giving integral recursive formulas for the vector space isomorphism between the Clifford algebra and the exterior algebra (in both directions). First, let us define everything in maximal generality: