Random generation of the special linear Lie algebra over finite fields

We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity, provided that $(n,p) \neq (3,3), (4,2)$.