An example of a discrete, abellian and not cyclic group in $PSL(2,\mathbb{C})$

Question

There is a theorem which says that: if $\Gamma$ is a abelian discrete subgroup of $PSL(2,\mathbb{R})$, then $\Gamma$ is cyclic. Nevertheless, we do not get it if the group is $PSL(2,\mathbb{C})$, I mean, there is an abelian discrete subgroup which are not cyclic. The problem is that I have not managed to figure out which it is.

Answer 1

A simple example is given by the matrices $$\pmatrix{1&a+bi\\0&1}$$ for $a$, $b\in\Bbb Z$.

Answer 2

For a classification of abelian discrete subgroups of $PSL(2,\Bbb C)$ see the book Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory by Elstrodt, Grunewald and Mennicke, Theorem $1.8$ and the remark before Corollary $1.9$ on page $38-39$.