Conjugation of Quaternions as Rotations in $\mathbb{R}^3$

Question

In a lot of published texts, when I read about the conjugacy class of pure quaternions being rotations, they assume that the norm of the non-pure quaternion is 1. Is there a reason that this is necessary? Doesn't the conjugacy map preserve length of the pure quaternion it's rotating no matter what since the norm of a quaternion is the reciprocal of the norm of its inverse? Or is it just that this assumption about the norm gives some nicer looking resultant pure quaternion?

Answer 1

Leaving anon's comment as a CW answer to get this off the list of unanswered questions.

If $r$ is a non-zero real number, it makes no difference whether you conjugate with $w$ a or $wr$. This is because all the quaternions commute with $r$ so $$ wqw^{-1}=wqrr^{-1}w^{-1}=wrqr^{-1}w^{-1}=(wr)q(wr)^{-1}. $$ Therefore we are at liberty to conjugate with $w/\|w\|$ instead of $w$, and may assume that $\|w\|=1$.