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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 $\Big(\frac{-1,-1}{\mathbb{Q}}\Big)$?

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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.

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    Further, I think Wiki's description in this case is ... unfortunate. In my experience, the "Hurwitz order" or whatever refers to the first thing in the question. The second thing may indeed have been investigated by Hurwitz (I don't know), and does indeed fit into a general understanding (e.g., see Weil's "Basic Number theory", or my notes on such stuff), but there's no immediate connection between those two things... apart from being two instances of a more general thing. – 2017-01-22
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    @paulgarrett I agree the nomenclature is unfortunate, but this time it is not Wikipedia's fault. See my answer. – 2017-01-22
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What they have to do is they are both related to things studied by Hurwitz. The "Hurwitz quaternions" are an object typically considered in number theory (and as a number theorist, if I hear "Hurwitz order" that's what I'll think of), and were directly studied by Hurwitz.

The other "Hurwitz order" comes up in geometry because it is related to Hurwitz surfaces, which are compact Riemann surfaces that have $84(g-1)$ automorphisms ($g =$ genus). This is the maximal number possible by a theorem of Hurwitz. The smallest possible genera for Hurwitz surfaces are 3, 7 and 14. The first two have automorphism groups of the form PSL(2,$q$), whereas the third has automorphism group the triangle group (2,3,7). The latter group is essentially the unit group in the other "Hurwitz order," hence the name, though as far as I know was not actually studied by Hurwitz.