Two-generation of traceless matrices over finite fields

We prove that the Lie algebra ${\mathfrak{s}\mathfrak{l}}_n({\mathbf{\text{F}}}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n,p)$ is $(3,3)$ or $(4,2)$. In the latter cases, we establish curious identities that obstruct $2$-generation.