Remark. why the Hurwitz order is introduced [fgap-0017]

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Hurwitz developed integral quaternions in 1919. Starting from the Lipschitz order \(L\), the question is not merely which quaternions have integral coordinates, but which order has the better arithmetic structure. The Lipschitz order is not maximal. Voight compares this with enlarging \(\mathbb {Z}[\sqrt {-3}]\) to the Eisenstein integers. Since \(a=i+j+k\) satisfies \(a^2=-3\), the analogous element \[ \omega =\frac {-1+a}{2} \] satisfies \(\omega ^2+\omega +1=0\). Adjoining it enlarges \(L\) to the maximal Hurwitz order \(\mathcal {O}\). See [voight2021quaternion, sec. 11.1, pp. 165--166].

This enlargement supplies more than a convenient lattice. Its \(24\) units form the binary tetrahedral group, and conjugation by \(\omega \) cyclically permutes the quaternion units \(i,j,k\). The same order also supports a norm-Euclidean algorithm. These features make the Hurwitz order a small meeting point of quaternion arithmetic, finite-group structure, and explicit calculation; see [voight2021quaternion, secs. 11.2--11.3, pp. 166--169].