  We produce new short laws in two variables valid in finite groups of Lie
type. Our result improves upon results of Kozma and the second named author,
and is sharp up to logarithmic factors, for all families except possibly the
Suzuki groups. We also produce short laws valid for generating pairs and random
pairs in finite groups of Lie type, and, conditional on Babai's diameter
conjecture, make effective the dependence of our bounds on the rank. Our proof
uses, among other tools, the Classification of Finite Simple Groups,
Aschbacher's structure theorem for maximal subgroups for classical groups, and
upper bounds on the diameters of finite simple groups due to Breuillard, Green,
Guralnick, Pyber, Szabo and Tao.
