= Solution
Identify the <Möbius group> with $\operatorname{PSL}_2(\mathbb C)$, the orientation-preserving <isometry group> of three-dimensional <hyperbolic space>. Here a <properly discontinuous group action> permits finite stabilizers: for each compact set $K$, only finitely many $g$ satisfy $gK\cap K\ne\varnothing$.
Fix $o\in\mathbb H^3$. The set of isometries with $d(o,go)\leq R$ is compact. To see this, an orientation-preserving isometry is uniquely determined by the image of an oriented orthonormal frame at $o$. The image point lies in a compact closed hyperbolic ball and its frame lies in a compact rotation-group fiber. These choices parametrize the indicated set of isometries continuously.
A <discrete subgroup> of a <topological group> is closed: otherwise distinct subgroup elements accumulating at the same point would have quotients tending to the identity, contradicting isolation of the identity. Its intersection with a compact set is therefore finite. If $K\subseteq B(o,R)$ and $gK\cap K\ne\varnothing$, choosing $x,gx\in K$ gives $d(o,go)\leq d(o,gx)+d(gx,go)\leq2R$. Hence discreteness implies a <properly discontinuous group action>. Conversely, if the subgroup is not discrete, distinct $g_n$ tend to the identity. For a compact ball $K$ with $o$ in its interior, eventually $g_no\in K$, so infinitely many $g_nK$ meet $K$, contradicting proper discontinuity. This proves the equivalence.
Matrices of determinant one over the <Gaussian integers> are closed under multiplication and inversion, with inverse $\left(\begin{smallmatrix}d&-b\\-c&a\end{smallmatrix}\right)$. Quotienting by $\{\pm I\}$ gives the specified group of <Möbius transformations>. The Gaussian integers form a <discrete set> in the complex plane, so these matrices form a <discrete subgroup> of $\operatorname{SL}_2(\mathbb C)$, and the finite quotient remains discrete. Thus this group acts properly discontinuously on $\mathbb H^3$.
Explicit representatives of all four types are
$$
\boxed{
\begin{array}{c|c|c}
\text{type}&\text{matrix}&\text{trace}\\\hline
\text{elliptic}&\begin{pmatrix}0&-1\\1&0\end{pmatrix}&0\\
\text{parabolic}&\begin{pmatrix}1&1\\0&1\end{pmatrix}&2\\
\text{hyperbolic}&\begin{pmatrix}2&1\\1&1\end{pmatrix}&3\\
\text{loxodromic}&\begin{pmatrix}1+i&1\\i&1\end{pmatrix}&2+i
\end{array}}
$$
The first is $z\mapsto-1/z$ and has order two; the second is the nonidentity translation $z\mapsto z+1$. The third has positive real reciprocal <eigenvalues> $(3\pm\sqrt5)/2$, giving translation along an axis without rotation. For the last, a nonreal trace excludes both unit-modulus and real eigenvalues. Its multiplier therefore has modulus different from one and a nonzero rotation angle: it is a <loxodromic Möbius transformation>.
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