are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Each of the algebras and is isomorphic to or , where is a full matrix ring with entries from , , or . For a complete classification of these algebras see Classification of Clifford algebras. Real numbers 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: where is the dimension of the vector space. The pair of integers is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted The Clifford algebra on is denoted The symbol means either or depending on whether the author prefers positive-definite or negative-definite spaces. A standard basis for consists of mutually orthogonal vectors, of which square to +1 and of which square to −1. Of such a basis, the algebra will therefore have vectors that square to +1 and vectors that square to −1. A few low-dimensional cases are: is naturally isomorphic to since there are no nonzero vectors. is a two-dimensional algebra generated by that squares to −1, and is algebra-isomorphic to , the field of complex numbers. is a four-dimensional algebra spanned by The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the quaternions is an 8-dimensional algebra isomorphic to the direct sum , the split-biquaternions. Complex numbers 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 Thus, for each dimension , up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on with the standard quadratic form by . For the first few cases one finds that , the complex numbers , the bicomplex numbers , the biquaternions where denotes the algebra of matrices over . Examples: constructing quaternions and dual quaternions Quaternions In this section, Hamilton's quaternions are constructed as the even subalgebra of the Clifford algebra Let the vector space be real three-dimensional space and the quadratic form be the usual quadratic form. Then, for in we have the bilinear form (or scalar product) Now introduce the Clifford product of vectors