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/Tr

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 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.