Diameter bounds for finite simple Lie algebras

We prove strong and explicit diameter bounds for finite simple Lie algebras, which parallel Babai's conjecture for finite simple groups. Specifically, we show that any nonabelian finite simple Lie algebra $\mathfrak{g}$ over $\mathbf{F}_p$ has diameter $O((\log |\mathfrak{g}|)^D)$ for $D \approx 3.11$ with respect to any generating set. For absolutely simple classical Lie algebras over $\mathbf{F}_p$, we establish the sharper bound $O(\log |\mathfrak{g}|)$ when the Lie type is fixed and the generators are chosen uniformly at random.