What do the Hurwitz quaternions have to do with the Hurwitz quaternion order? The Hurwitz quaternions are the ring formed by the elements of the form $w+xi+yj+zij$ where $i^2=j^2=-1$, $ij=-ji$, and where $w,x,y,z$ are either all integers or all half-integers. These form a maximal order of the quaternion algebra $\Big(\frac{-1,-1}{\mathbb{Q}}\Big)$.
The Hurwitz quaternion order, on the other hand, is defined as follows (according to Wikipedia). Let $\rho$ be the primitive seventh root of unity and let $K$ be the maximal real subfield of $\mathbb{Q}(\rho)$. Let $\eta=2\cos(\frac{2\pi}{7})$ (so that $\mathbb{Z}[\eta]$ is the ring of integers of $K$) and consider the quaternion algebra $\Big(\frac{\eta,\eta}{K}\Big)$ (where $i^2=j^2=\eta$). Then let $\tau=1+\eta+\eta^2$ and $j'=\frac{1+\eta i+\tau j}{2}$, and the Hurwitz quaternion order is the maximal order $\mathbb{Z}[\eta][i,j,j']$ in $\Big(\frac{\eta,\eta}{K}\Big)$.
It seems that the Hurwitz quaternion order should be some sort of generalization of the Hurwitz quaternions but there are a lot of decisions here that seem arbitrary to me. What is the motivation for the similar nomenclature? What is special about the order $\mathbb{Z}[\eta][i,j,j']$ in $\Big(\frac{\eta,\eta}{K}\Big)$ and what does it have in common with the Hurwitz quaternions in $\Big(\frac{-1,-1}{\mathbb{Q}}\Big)$?

It appears that the term does not refer to a generalization of the relationship of Hamilton's quaternions to the Hurwitz quaternions (as one might expect) but rather the term as defined there is just a specific order in a specific quaternion algebra other than $\mathbb H$.  Go figure.
