Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$, and it is finitely generated by some known generators. That is, $G=\langle g_1,\dots,g_n\rangle$.

The Frobenius norm of a matrix $m=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ is $\|m\|:=\sqrt{|a|^2+|b|^2+|c|^2+|d|^2}$. The set of elements in $G$ having the same Frobenius norm is a discrete subset of a compact set, and therefore is finite.

I'm interested in algorithms for enumerating elements of $G$ by their Frobenius norm. That is, find all elements of smallest (nontrivial) Frobenius norm, then find all elements of next smallest Frobenius norm, etc. I have different techniques for different cases using other properties of the groups, but what is an efficient algorithm to do this generally, knowing only that we are given the generators?