Universal differentiability sets and maximal directional derivatives in Carnot groups

We show that every Carnot group Gof step 2 admits a Hausdorff dimension one ‘universal differentiability set’ Nsuch that every Lipschitz map f:G →Ris Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of fat a point ximplies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.