Clifford algebra

When the characteristic of the field is not 2, this may be replaced by what is then an equivalent requirement,

The fundamental Clifford identity implies that for an orthogonal basis

Clifford algebras are also sometimes referred to as geometric algebras, most often over the real numbers.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension n is equivalent to the standard diagonal form

In this section we assume that characteristic is not 2, the vector space is finite-dimensional and that the associated symmetric bilinear form of

Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.

To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)