What is "split-quaternion"

Coquaternion multiplication

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In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempotents. (For example, (1 + j) is an idempotent zero-divisor, and i − j is nilpotent.) As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below.

The set {1, i, j, k} forms a basis. The products of these elements are

ij = k = −ji, jk = −i = −kj, ki = j = −ik, i = −1, j = +1, k = +1,

and hence ijk = 1. It follows from the defining relations that the set {1, i, j, k, −1, −i, −j, −k} is a group under coquaternion multiplication; it is isomorphic to the dihedral group of a square.

A coquaternion

q = w + xi + yj + zk, has a conjugate q* = w − xi − yj − zk.

Due to the anti-commutative property of its basis vectors, the product of a coquaternion with its conjugate is given by an isotropic quadratic form:

N(q) = qq = w + x − y − z.
Given two coquaterions p and q, one has N(p q) = N(p) N(q), showing that N is a quadratic form admitting composition. This algebra is a composition algebra and N is its norm. Any q ≠ 0 such that N(q) = 0 is a null vector, and its presence means that coquaternions form a "split composition algebra", and hence a coquaternion is also called a split quaternion.

When the norm is non-zero, then q has a multiplicative inverse, namely q*/N(q). The set

U = {q : qq* ≠ 0}

is the set of units. The set P of all coquaternions forms a ring (P, +, •) with group of units (U, •). The coquaternions with N(q) = 1 form a non-compact topological group SU(1,1), shown below to be isomorphic to SL(2, R).

Historically coquaternions preceded Cayley's matrix algebra; coquaternions (along with quaternions and tessarines) evoked the broader linear algebra.