Biquaternion

In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

• Biquaternions when the coefficients are complex numbers.
• Split-biquaternions when the coefficients are split-complex numbers.
• Dual quaternions when the coefficients are dual numbers.

This article is about the ordinary biquaternions named by William Rowan Hamilton in 1844.^[1] 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 representation of the Lorentz group, which is the foundation of special relativity.

The algebra of biquaternions can be considered as a tensor product C ⊗_R H, where C is the field of complex numbers and H is the division algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M₂(C). They are also isomorphic to several Clifford algebras including C ⊗_R H = Cl^[0]
₃(C) = Cl₂(C) = Cl_1,2(R),^[2] the Pauli algebra Cl_3,0(R),^[3][4] and the even part Cl^[0]
_1,3(R) = Cl^[0]
_3,1(R) of the spacetime algebra.^[5]