A wonderful explanation of why is rotation of
, where
is most easily interpreted with the complex-number like structure of
with
normalized (i.e.,
). Note that
is then just
(since
just like the imaginary
for complex number).
Basically quarternion is 4D stereographic projection back to 3D space. When multiply by a quarternion from the left, points parallel to
will be translate along $\bf v$ and points penpendicular to
will rotate around
(following the right hand rule). Similarly, when multiplying from the right by the same quarternion, points parallel to
will be translated in the same direction, but points perpendicular to
will be rotated in the opposite direction (following left hand rule). So if we multiple
and
from both left and right hand sides, the translation effect will be exactly cancelled, and the the rotation will be doubled. That is why the angle in
should be halved.