Unit quaternion composition visualization
ij=k example:
The first transvection pair j = (0,0,1,0) reflects in X then -Z or any
equivalent pair (u1,u2) obtained by rotating this pair in the plane it
spans, so ( (u1,u2) u1 x u2 ) = (0,0,1,0). The unit quaternion j acts on R^3
by rotating by pi (twice the angle between the transvection pair) about Y,
consistent with the usual action by conjugation on pure vector quaternions,
v->-jvj. The first transvection pair i = (0,1,0,0) reflects in -Z then Y or any
equivalent pair (u1,u2) obtained by rotating this pair in the plane it
spans, so ( (u1,u2) u1 x u2 ) = (0,1,0,0). The unit quaternion i acts on R^3
by rotating by pi (twice the angle between the transvection pair) about X.
The composition is obtained as a transvection pair by choosing the special
representative pairs given for ij (j followed by i) for which the second
vector of the first pair and the first vector of the second pair agree, and cancelling
the identical pair of reflections "in the middle". This leaves
reflection in -X then Y, and ( (u1,u2) u1 x u2 ) = (0,0,0,1)=k.
The unit quaternion k acts on R^3 by rotating by pi (twice the angle between
the transvection pair) about Z. The order of the pairs is indicated by green
for initial and red for final. The viewpoint is (5,5,5). The rotation
of the pairs merely displays their equivalence classes, not to be
confused with the corresponding rotations.