Fundamental polyhedra of projective elementary groups

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.