3 November 2011 UTC Whoa I really lost you here The affine space was only an analogy I do not want to construct something new only show that there are two classes in common usage Clifford ring my name and Clifford algebra furthermore that some inaccuracy occurs because they are not adequately distinguished All I m saying is that there exists a class of objects Clifford ring in which no specific subspace is necessarily known as the generating vector space V You can hide the identity of the subspace V and the grading by mysterious relabelling But here is the point this relabelling truly does hide which of the basis vectors span V it is possible to form a product for every element in the ring but impossible to identify the original subspace V used in the construction The quaternions are a ring isomorphic to C 0 2 which has a 1 2 1 grading structure Yet given can you find a unique decomposition with this grading structure No In a CA grading specifically refers to a unique natural decomposition into I think it is into a direct sum of 0 vectors 1 vectors 2 vectors etc talkcontr 19 22