hyperbolic versor corresponds to a velocity in direction of speed where is the velocity of light. The inertial frame of reference of this velocity can be made the resting frame by applying the Lorentz boost given by since then so that Naturally the hyperboloid which represents the range of velocities for sub-luminal motion, is of physical interest. There has been considerable work associating this "velocity space" with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group provides a group representation for the Lorentz group. After the introduction of spinor theory, particularly in the hands of Wolfgang Pauli and Élie Cartan, the biquaternion representation of the Lorentz group was superseded. The new methods were founded on basis vectors in the set which is called the complex light cone. The above representation of the Lorentz group coincides with what physicists refer to as four-vectors. Beyond four-vectors, the standard model of particle physics also includes other Lorentz representations, known as scalars, and the -representation associated with e.g. the electromagnetic field tensor. Furthermore, particle physics makes use of the representations (or projective representations of the Lorentz group) known as left- and right-handed Weyl spinors, Majorana spinors, and Dirac spinors. It is known that each of these seven representations can be constructed as invariant subspaces within the biquaternions. As a composition algebra Although W.R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quaternions out of complex numbers in the so-called Cayley–Dickson construction. In this construction, a bicomplex number (w,z) has conjugate (w,z)* = (w, – z). The biquaternion is then a pair of bicomplex numbers (a,b), where the product with a second biquaternion (c, d) is If then the biconjugate When (a,b)* is written as a 4-vector of ordinary complex numbers, The biquaternions form an example of a quaternion algebra, and it has norm Two biquaternions p and q satisfy indicating that N is a quadratic form admitting composition, so that the biquaternions form a composition algebra. See also Biquaternion algebra Hypercomplex number Joachim Lambek MacFarlane's use Quotient ring Notes References Arthur Buchheim (1885) "A Memoir on biquaternions",