This book is only an introduction to Clifford algebras. Here are some suggestions for further reading; they are not meant to provide a comprehensive bibliography, but rather to indicate where to go next.

The algebraic environment

Further results about multilinear mappings and tensor products, are given in standard textbooks, such as Cohn [Coh], Jacobson [Jac] and Mac Lane and Birkhoff [MaB]. The results are presented in the more general setting of modules over a commutative ring. This leads to serious problems which do not arise in the vector space case. The idea of considering representations of algebras in terms of modules extends to infinite-dimensional algebras, such as C*-algebras. A good starting point for this is the book by Lance [Lan].

The proof of the existence of the tensor product of two modules over a commutative ring, as described in the remark at the end of Section 3.2, does not lead to a simple description of the structure of the tensor product, and there are also problems with torsion. Tensor products of vector spaces are so much more straightforward that they deserve to be treated separately.

Quadratic spaces

Lam [Lam] is the standard work on quadratic forms, and is a goldmine of mathematics. It considers quadratic forms over fields not of characteristic 2.