and given by This formulation uses the negative sign so the correspondence with quaternions is easily shown. Denote a set of orthogonal unit vectors of as then the Clifford product yields the relations and The general element of the Clifford algebra is given by The linear combination of the even degree elements of defines the even subalgebra with the general element The basis elements can be identified with the quaternion basis elements as which shows that the even subalgebra is Hamilton's real quaternion algebra. To see this, compute and Finally, Dual quaternions In this section, dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form. Let the vector space be real four-dimensional space and let the quadratic form be a degenerate form derived from the Euclidean metric on For in introduce the degenerate bilinear form This degenerate scalar product projects distance measurements in onto the hyperplane. The Clifford product of vectors and is given by Note the negative sign is introduced to simplify the correspondence with quaternions. Denote a set of mutually orthogonal unit vectors of as then the Clifford product yields the relations and The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element The basis elements can be identified with the quaternion basis elements and the dual unit as This provides the correspondence of with dual quaternion algebra. To see this, compute and The exchanges of and alternate signs an even number of times, and show the dual unit commutes with the quaternion basis elements Examples: in small dimension Let be any field of characteristic not . Dimension 1 For , if has diagonalization , that is there is a non-zero vector such that , then is algebra-isomorphic to a -algebra generated by an element satisfying , the quadratic algebra . In particular, if (that is, is the zero quadratic form) then is algebra-isomorphic to the dual numbers algebra over . If is a non-zero square in , then . Otherwise, is isomorphic to the quadratic field extension of . Dimension 2 For , if has diagonalization with non-zero and (which always exists if is non-degenerate), then is isomorphic to a -algebra generated by elements and satisfying , and . Thus is isomorphic to the (generalized) quaternion algebra . We retrieve Hamilton's quaternions when