Title: Two-generation of traceless matrices over finite fields

Abstract: We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of odd characteristic $p$ can be generated by $2$ elements with a single exception $(n, p)= (3, 3)$. We establish a curious trace identity in $\mathfrak{sl}_3(\textbf{F})$ over any field $\textbf{F}$ of characteristic $3$ that obstructs $2$-generation.