What is "biquaternion"

n. (context algebra English) Any of the numbers $w\; +\; xi\; +\; yj\; +\; zk\; !$ where ''w'', ''x'', ''y'', and ''z'' are complex numbers and the elements of {1, ''i'', ''j'', ''k''} multiply as in the quaternion group.

In abstract algebra, the biquaternions are the numbers , where w, x, y, and z are complex numbers and the elements of multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion:

• (Ordinary) biquaternions when the coefficients are (ordinary) complex numbers
• Split-biquaternions when w, x, y, and z are split-complex numbers
• Dual quaternions when w, x, y, and z are dual numbers.

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844 (see Proceedings of Royal Irish Academy 1844 & 1850 page 388). Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a presentation of the Lorentz group, which is the foundation of special relativity.

The algebra of biquaternions can be considered as a tensor product (taken over the reals) where C is the field of complex numbers and H is the algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the (real) quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2×2 complex matrices M(C). They can be classified as the Clifford algebra . This is also isomorphic to the Pauli algebra Cℓ(R), and the even part of the spacetime algebra Cℓ(R).