To see this, compute and Finally,  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 "V" be real four-dimensional space R4, and let the quadratic form "Q" be a degenerate form derived from the Euclidean metric on R3. For "v", "w" in R4 introduce the degenerate bilinear form This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane. The Clifford product of vectors "v" and "w" is given by Note the negative sign is introduced to simplify the correspondence with quaternions. Denote a set of mutually orthogonal unit vectors of R4 as "e"1, "e"2, "e"3 and "e"4, 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 "i", "j", "k" and the dual unit "ε" as This provides the correspondence of Cl(R) with dual quaternion algebra. To see this, compute and The exchanges of "e"1 and "e"4 alternate signs an even number of times, and show the dual unit "ε" commutes with the quaternion basis elements "i", "j", and "k". Let "K" be any field of characteristic not 2. For , if "Q" has diagonalization diag("a"), that is there is a non-zero vector "x" such that , then is algebra-isomorphic to a "K"-algebra generated by an element "x" satisfying , the quadratic algebra . In particular, if (that is, "Q" is the zero quadratic form) then is algebra-isomorphic to the dual numbers algebra over "K". If "a" is a non-zero square in "K", then . Otherwise, is isomorphic to the quadratic field extension "K"() of "K". For , if "Q" has diagonalization with non-zero "a" and "b" (which always exists if "Q" is non-degenerate), then is isomorphic to a "K"-algebra generated by elements "x" and "y" satisfying , and .