Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the congruence subgroup defined by $\alpha$, i.e. the kernel of the natural map $\text{SL}_n(\mathcal{O}) \rightarrow \text{SL}_n(\mathcal{O}/\alpha)$. I am interested in finding generators for $\text{SL}_n(\mathcal{O},\alpha)$.

This was ``almost'' done by Bass-Milnor-Serre, who proved the following. Letting $\text{E}_n(\mathcal{O},\alpha)$ be the *normal* subgroup of $\text{SL}_n(\mathcal{O},\alpha)$ generated by elementary matrices whose off-diagonal entries lie in $\alpha$, we have

- The group $\text{Q}_n(\mathcal{O},\alpha) := \text{SL}_n(\mathcal{O},\alpha) / E_n(\mathcal{O},\alpha)$ is a finite cyclic group which is isomorphic to a subgroup of the roots of unity lying in $k$.
- If $k$ has a real embedding, then $\text{Q}_n(\mathcal{O},\alpha)=0$.

The only remaining problem, then, is when $k$ does not have a real embedding, so when $k$ is totally imaginary. My question is then in that case, does anyone know an explicit matrix (or matrices) in $\mathcal{SL}_n(\mathcal{O},\alpha)$ which projects to an element (or elements) which generates $\text{Q}_n(\mathcal{O},\alpha)$? Bass-Milnor-Serre also show that we may take said matrices to lie in $\text{SL}_2(\mathcal{O})$, but I do not demand this.