Hurwitz quaternions are defined as:
$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$ (that is, all integer or half integer quaternions).
I'm trying to show that if $a$ is a unit, then $||a||=1$ (the usual quaternion norm), but I can only sort of show that using very tedious calculations. Is there a short(er) and more elegant way to see this, preferably purely algebraic?