Hurwitz quaternion order

Example

Let $\eta = 2\cos(2\pi/7)$ and let $K = \Q(\eta)$, the real cubic subfield of the seventh cyclotomic field, an algebraic number field with ring of integers $\Z[\eta]$. The number $\eta$ is a root of $t^3 + t^2 - 2t - 1$. Let $D$ be the quaternion algebra $(\eta, \eta)_K$, with $K$-basis $1, i, j, ij$ and relations $i^2 = j^2 = \eta$ and $ij = -ji$. Put $\tau = 1 + \eta + \eta^2$ and $j' = \tfrac{1}{2}(1 + \eta i + \tau j)$. The Hurwitz quaternion order is the $\Z[\eta]$-order $$\mathcal{Q}_{\mathrm{Hur}} = \Z[\eta][i, j, j']$$ generated by $i$, $j$ and $j'$. It is a maximal order of $D$.

A check that $j'$ is integral: an element $a + bi + cj + d\,ij$ of $D$ has reduced trace $2a$ and reduced norm $a^2 - \eta b^2 - \eta c^2 + \eta^2 d^2$. So $j'$ has reduced trace $1$ and reduced norm $\tfrac{1}{4}(1 - \eta^3 - \eta\tau^2)$. Using $\eta^3 = -\eta^2 + 2\eta + 1$ one finds $\eta \tau^2 = \eta^2 + 10\eta + 4$, so the reduced norm is $\tfrac{1}{4}(-4 - 12\eta) = -1 - 3\eta \in \Z[\eta]$. Hence $j'$ satisfies the monic equation $x^2 - x - 1 - 3\eta = 0$ over $\Z[\eta]$.