We prove that the Lie algebra slₙ(Fq) 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 sl₃(F) over any field F of characteristic $3$ that obstructs $2$-generation.
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Jezernik et al. (2024) studied this question.
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