Gaussian-integer disc group with an ideal-square Dirichlet domain ($D=\{2(|x|+|y|)\le 1+x^2+y^2\}\cap \mathbb{D}$)

Consider determinant-one matrices $\left(\begin{array}{cc}u& v\\ \overline{v}& \overline{u}\end{array}\right)$ over Gaussian integers with $u+v\equiv 1(\mathrm{mod}2\mathbb{Z}[i])$, modulo their common sign. They act on the Poincare disc model by fractional linear maps. Their origin-centered Dirichlet domain isFor a nonidentity matrix set $N=|v{|}^2\ge 1$ and $uv=m+in$. The group orbit point is $(m+in)/(N+1)$ and its bisector inequality is $2(mx+ny)\le N(1+x^2+y^2)$. Since $m^2+n^2=N(N+1)$, integrality gives $|m|,|n|\le N$, so the four inequalities defining $D$ imply all of them. They are necessary because the matrices with $u=1\pm i$, $v=\pm i$ realize the four combinations $m,n=\pm 1$. The four sides are arcs of the unit-radius circles centered at $\pm 1\pm i$, and the ideal vertices are $1,i,-1,-i$.