Split-quaternion

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 nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(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

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.

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