# How to enumerate a discrete group of matrices by their Frobenius norm?

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?