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A Kleinian group is a discrete subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$, acting as orientation-preserving hyperbolic isometries of ${\mathbb{H}}^3$ and as Möbius transformations of its boundary sphere. Its limit set consists of the accumulation points on the boundary of an orbit $Gx$ with $x\in {\mathbb{H}}^3$; it is independent of that interior base point and is closed. A nonidentity parabolic transformation is conjugate to $z\mapsto z+c$, $c\ne 0$. Its iterates on the upper half-space tend to its unique fixed boundary point, infinity in these coordinates. Consequently every parabolic fixed point belongs to the limit set.

The displayed matrix has determinant one, trace two, and is not the identity because $a\ne 0$. Its fixed-point equation is $a(z-w{)}^2=0$, so it has exactly the double fixed point $w$. It is therefore parabolic, and its fixed point is $w$.

The determinant-one Gaussian-integer matrices form a subgroup of ${\mathrm{SL}}_2(\mathbb{C})$. They are discrete because their entries lie in a discrete lattice: a matrix sufficiently near the identity has every entry equal to the corresponding identity entry. Passing through the quotient by $\{\pm I\}$ preserves discreteness, so their Möbius transformations form a Kleinian group.

For $w=n/r$, $n=p+iq$ and $r\ne 0$, set $a=r^2$. This gives the Gaussian-integer parabolic constructionAll entries lie in $\mathbb{Z}[i]$, the determinant is one, and its parabolic fixed point is $n/r=w$. The points with nonzero $p,q,r$ are dense in the finite complex plane, and their closure on the sphere includes infinity. Since the limit set is closed and contains all these parabolic fixed points, the limit set is the whole Riemann sphere.