Definition. the Hurwitz order and its units [fgap-0016]

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Inside the rational quaternion algebra \(B=(-1,-1\mid \mathbb {Q})\), let \(k=ij\) and \[ \omega =\frac {-1+i+j+k}{2}. \] The Lipschitz order and Hurwitz order are the \(\mathbb {Z}\)-lattices \[ \begin {aligned} L&=\mathbb {Z}+\mathbb {Z}i+\mathbb {Z}j+\mathbb {Z}k,\\ \mathcal {O} &=\mathbb {Z}+\mathbb {Z}i+\mathbb {Z}j+\mathbb {Z}\omega . \end {aligned} \] Both are subrings containing \(1\). The order \(\mathcal {O}\) contains \(L\) with index \(2\); it is the unique order properly containing \(L\), and it is maximal. See [voight2021quaternion, sec. 11.1, esp. lem. 11.1.2].

A Hurwitz unit is a unit of \(\mathcal {O}\). Equivalently, it is an element \(q\in \mathcal {O}\) whose reduced norm is \(1\). Voight calculates that \(\mathcal {O}^{\times }\) has \(24\) elements in [voight2021quaternion, sec. 11.2, pp. 166--168]. The particular unit \(\omega \) will give the cyclic factor used below.