在抽象代数中，分裂四元数（split-quaternions）或反四元数（coquaternions）是一种四维的结合代数的元素，由James Cockle（英语：James Cockle）在1849年引入，当时称为反四元数。 类似于汉密尔顿1843年引入的四元数 ，它们组成了一个四维的实向量空间,且有乘法运算。 与四元数不同，分裂四元数包含非平凡的零因子、幂零元和幂等元（英语：Idempotent_(ring_theory)）。（例如， ${\displaystyle \frac{1}{2}(1+j)}$ 是其范数。 任何满足${\displaystyle q\ne <!--\; \ne \; -->0}$有倒数，即 $\frac{{\displaystyle q^{\ast <!--\; \ast \; -->}}}{N(q)}$表示为矩阵环，其中的分裂四元数的乘法与矩阵乘法的行为相同。例如，这个矩阵的行列式是

减号的出现将反四元数与使用了加号的四元数${\displaystyle \mathbb{H}}$和是双曲复数，分裂四元数${\displaystyle q=(a,b)=((w+zj),(y+xj))}$∗ is positive definite on the planes and . Consider the counter-sphere {: ∗ = −1}.

Take = + i + where = j cos() + k sin(). Fix and suppose

Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane ⊂ P, can be written, for some ∈

Let φ be the angle between the hyperbolas from to and . This angle can be viewed, in the plane tangent to the counter-sphere at , by projection:

as in the expression of angle of parallelism in the hyperbolic plane H2 . The parameter determining the meridian varies over the 1. Thus the counter-sphere appears as the manifold 1 × H2.

By using the foundations given above, one can show that the mapping

is an ordinary or hyperbolic rotation according as

The collection of these mappings bears some relation to the Lorentz group since it is also composed of ordinary and hyperbolic rotations. Among the peculiarities of this approach to relativistic kinematic is the anisotropic profile, say as compared to hyperbolic quaternions.

Reluctance to use coquaternions for kinematic models may stem from the (2, 2) signature when spacetime is presumed to have signature (1, 3) or (3, 1). Nevertheless, a transparently relativistic kinematics appears when a point of the counter-sphere is used to represent an inertial frame of reference. Indeed, if ∗ = −1, then there is a = i sinh() + cosh() ∈ such that ∈ , and a ∈ such that = exp(). Then if = exp(), = i cosh() + sinh(), and = i, the set {, , , } is a pan-orthogonal basis stemming from , and the orthogonalities persist through applications of the ordinary or hyperbolic rotations.

The coquaternions were initially introduced (under that name) in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 of the Quaternion Society. Alexander Macfarlane called the structure of coquaternion vectors an when he was speaking at the International Congress of Mathematicians in Paris in 1900.

The unit sphere was considered in 1910 by Hans Beck. For example, the dihedral group appears on page 419. The coquaternion structure has also been mentioned briefly in the .