It is well known that a nonabelian finite simple group, say $\mathrm{PSL}_n(\mathbf{F}_p)$, can be generated by two elements. In fact, the probability that two elements generate it tends to $1$ as the size of the group tends to infinity.
Is the same result about probabilistic generation known to hold for Lie algebras over finite fields, say $\mathfrak{sl}_n(\mathbf{F}_p)$?