 Gaussian-integer parabolic construction

For $n\in \mathbb{Z}[i]$ and a nonzero integer $r$, the displayed determinant-one matrix has trace two and represents a nonidentity parabolic Möbius transformation fixing $n/r$. Its fixed-point equation is $(rz-n{)}^2=0$. These parabolic fixed points are dense in the Riemann sphere, proving that the Gaussian-integer determinant-one Kleinian group has full limit set.