[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.
[5] Let {1, i, j, k} be the basis for the (real) quaternions H, and let u, v, w, x be complex numbers, then is a biquaternion.
[6] To distinguish square roots of minus one in the biquaternions, Hamilton[7][8] and Arthur W. Conway used the convention of representing the square root of minus one in the scalar field C by h to avoid confusion with the i in the quaternion group.
The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, 1901 by Charles Jasper Joly, reduced the biquaternion coverage in favour of the real quaternions.
Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers C. The algebra of biquaternions is associative, but not commutative.
When this matrix product is interpreted as i j = k, then one obtains a subgroup of matrices that is isomorphic to the quaternion group.
The squares of the elements hi, hj, and hk are all positive one, for example, (hi)2 = h2i2 = (−1)(−1) = +1.
The subalgebra given by is ring isomorphic to the plane of split-complex numbers, which has an algebraic structure built upon the unit hyperbola.
A third subalgebra called coquaternions is generated by hj and hk.
The linear subspace with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra.
In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M2(C) representation, are called Pauli matrices.
This allows the inverse to be defined by Consider now the linear subspace[10] M is not a subalgebra since it is not closed under products; for example
Proof: Note first that gg* = 1 implies that the sum of the squares of its four complex components is one.
Let r represent an element of the sphere of square roots of minus one in the real quaternion subalgebra H. Then (hr)2 = +1 and the plane of biquaternions given by
is a commutative subalgebra isomorphic to the plane of split-complex numbers.
Hence these algebraic operators on the hyperbola are called hyperbolic versors.
For every square root r of minus one in H, there is a one-parameter group in the biquaternions given by
The space of biquaternions has a natural topology through the Euclidean metric on 8-space.
Moreover, it has analytic structure making it a six-parameter Lie group.
When viewed in the matrix representation, G is called the special linear group SL(2,C) in M(2, C).
Many of the concepts of special relativity are illustrated through the biquaternion structures laid out.
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.
[11] 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 (1, 0) ⊕ (0, 1)-representation associated with e.g. the electromagnetic field tensor.
[12] 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.
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 N(pq) = N(p) N(q), indicating that N is a quadratic form admitting composition, so that the biquaternions form a composition algebra.