Hilbert-Haar coordinates and Miranda's theorem in Lie groups

We study the interior regularity of solutions to a class of quasilinear equations of non-degenerate p-Laplacian type on Lie groups that admit a system of Hilbert-Haar coordinates. These are coordinates with respect to which every linear function has zero symmetrized second order horizontal derivatives. All Carnot groups of rank two are in this category, as well as the Engel group, the Goursat type groups, and those general Carnot groups of step three for which the non-zero commutators of order three are linearly independent.