Spinors were first used under that name by physicists in the field of quantum mechanics. E. Cartan (see [1]) discovered a more geometrical setting for them, at least in the case that the scalar field was either the reals or the complexes. In 1954, C. Chevalley [2] gave the definitive treatment for the algebraic theory of spinors in all dimensions and for all characteristics. Physicists and mathematicians continue to find applications for spinors and their related groups, especially in the rarefied atmosphere of general relativity and superstring theory (see [4]).
Cartan also discovered a 'principle of triality', which is treated fully by Chevalley [2]. Unfortunately, Chevalley's book has proved too abstract for many mathematicians and physicists, with the consequence that the general theory has come to be developed and to be understood properly rather slowly. The volume by Crumeyrolle appears at an opportune moment and is very welcome, being written with the logical care that appeals to the mathematician and with notation and concrete applications in mind that should appeal to the physicist. The author has contributed significantly to the general subject, and much of his work appears in this book.
Let M denote a finite-dimensional vector space over the field K and q a hyperbolic quadratic form on M with associated bilinear form B(x, y) = q(x + y) -q(x) -q(y).
So there is a basis x~ ..... x,, Yl ..... y, of M such that q(xi) = q(Yi) = B(xi, x j) = B(y,, y j) = 0 and B(x,, yj) = 6ij, for 1 ~< i, j ~< r. Let N and P be the subspaces of M spanned by {xl ..... x,} and {Yx ..... y,}, respectively. So N and P are disjoint, maximal totally singular subspaces of M. Let S = ^ N be the exterior algebra over N. Recall that S is a linear space over K with a basis consisting of formal products as follows: for each sequence tr = (il ..... ip), where 1 ~< il < i2 < "'" < ip ~< r, there is a basis element ~(tr) = xil ^ xi~ ^ "'" ^ xip. An associative product is determined on S by linearly and