Definition:Hurwitz Quaternion

Definition
A Hurwitz quaternion is a quaternion whose components are all either:
 * integers

or:
 * integers plus a half (i.e. halves of odd integers.

The set $H$ of all Hurwitz quaternions can therefore be defined as:
 * $H = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: \left({a, b, c, d \in \Z}\right) \text { or } \left({a, b, c, d \in \Z + \dfrac 1 2}\right)}\right\}$