Interesting stuff. Maybe there is a connection with the use of quaternion pairs to represent rotations in 4-space (as in the Coxeter paper).
Using quaternion pairs for 4-space rotations relies on the simple fact that each quaternion represents a hyperplane in 4-space (https://en.wikipedia.org/wiki/Transformation_geometry).
Using single quaternions to represent rotations in 3-space is geometrically a different mechanism, which AFAIK doesn't have such a simple geometrical explanation.