linguistic:paraphrasing, structural:reformatting, structural:summarization, linguistic:translation
Relation to quaternions

A point in 4-dimensional space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion P = u + xi + yj + zk.

A left-isoclinic rotation is represented by left-multiplication by a unit quaternion Q = a + bi + cj + dk. In matrix-vector language this is

Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion Q = p + qi + rj + sk, which is in matrix-vector form

In the preceding section (#Isoclinic decomposition) it is shown how a general 4D rotation is split into left- and right-isoclinic factors.

In quaternion language Van Elfrinkhof's formula reads

or, in symbolic form,

According to the German mathematician Felix Klein this formula was already known to Cayley in 1854.

Quaternion multiplication is associative. Therefore,

which shows that left-isoclinic and right-isoclinic rotations commute.