These contents are due principally to the Wikipedia article.

A vector space commonly comes with a quadratic form especially if the space has a geometric flavor in which the quadratic form provides a metric for the space. A modern definition of a Clifford algebra is the smallest associative algebra C over a given vector space V over the reals with a quadratic form q where xx=q(x) for all x in V. Clifford studied algebras where the quadratic form was negative definite (x≠0 → q(x) < 0) but modern physics needs this easy generalization.

Here is a proof outline to show that the conventional definition of Clifford spaces is equivalent to the above abstract definition.

Sylvester’s ‘law of inertia’ tells us that two spaces with quadratic forms are isomorphic when the forms have the same signature. To find the signature of a quadratic form q you define the symmetric bilinear function b such that b(x, x) = q(x).
This can be done as b(x, y) = (q(x+y) − q(x−y))/4. (This step fails for fields of characteristic 2.) Then choose a basis {ej} for V such that b(ej, ek) = 0 when j≠k and b(ej, ej) = −1, 0 or 1. Sylvester shows how to do this and shows that the number of −1’s, 0’s and 1’s is determined solely by q and does not depend on the choice of basis. These numbers, expressed somehow, form the signature of q. When q(x) = 0 for all x then we have the Grassmann algebra with all 0’s which we ignore here. In fact we exclude any 0’s and require that b(ej, ej) = ±1. b(ej, ej) = −1 for the algebras that Clifford studied.

Let B be a basis for V with n vectors. There are 2n subsets of B. Let x be such a subset and multiply the elements of x together. These 2n products form a set of 2n vectors which form a basis for C, the Clifford algebra proper. When this subset is empty we have a new basis vector that spans a space that is naturally isomorphic with the field of V. Products of subsets with just one element collectively span a space which is naturally isomorphic with V. The literature generally conflates the field with its internal model, and also V with its internal model. This confused me for many months but we shall do the same henceforth.

This version of my Scheme code makes Clifford algebras based on indefinite quadratic forms.

Minkowski proposed the signature (− + + +) for the space time metric and space-time has not been the same since. Clifford’s geometric algebra described rotations of 3D Euclidean space with a subset of the Clifford algebra based on signature (− − −). His algebra was like quaternions except for including reflections. He knew the generalization to n dimensions for which there was no other known generalization besides the Clifford algebra. The concept of rotation depends on a concept of the length of a vector which is to be preserved by the rotation. The quadratic form is the square of a length and an element from the Clifford Group operates on a member of V to preserve its quadratic form and thus its length. The Clifford group is a subset of the Clifford algebra and if t is in the group then the operation of t on v is tvα(t−1). This operation leaves elements of V in V and preserves the quadratic form.

Pauli matrices are isomorphic to a Clifford algebra with signature (+ + +). Almost: (Dirac’s matrices are isomorphic to a Clifford algebra with signature (− + + +).) It is more complicated than that but perhaps it shouldn’t be.

I take a dangerous leap here which I need to justify. A Clifford algebra with signature (+ − − −) has as its ‘Clifford group’ elements that provide all of the isometries of a uniformly negatively curved space as described here.