  For $O$ an imaginary quadratic ring, we compute a fundamental polyhedron of
$\text{PE}_2(O)$, the projective elementary subgroup of $\text{PSL}_2(O)$. This
allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman.
Namely, we obtain a presentation for $\text{PE}_2(O)$, show that it has
infinite-index and is its own normalizer in $\text{PSL}_2(O)$, and split
$\text{PSL}_2(O)$ into a free product with amalgamation that has
$\text{PE}_2(O)$ as one of its factors.
