kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ring of 2×2 real matrices which Karzel and Kist cast as the kinematic algebras of the elliptic plane and hyperbolic plane respectively. See the "Motivation and Historical Review" at page 437 of Rings and Geometry, R. Kaya editor.

Some of the other hypercomplex systems in