The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be ** boundedly generated**, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of sets: $$ G = \langle g_1 \rangle \cdots \langle g_m \rangle. $$

Is there an elementary/accessible proof of this fact? I would like to have a reference to a proof.

How small can we take $m$ to be?

Is it possible to take $m = 8$?

Elementary expressions for unimodular matrices. They determine that for $G = \mathrm{SL}_n(\mathbb Z)$ ($n > 2$) you can take elementary matrices for all the $g_i$ and get an upper bound $m \le \frac12(3n^2-n) + 36$. $\endgroup$ – algorithmshark Sep 11 '15 at 16:49