  We study the sub-Riemannian structure determined by a left-invariant
distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the
conjectured cut times of Y. Sachkov for the sub-Riemannian Cartan problem.
Along the proof, we obtain a comparison with the known cut times in the
sub-Riemannian Engel group, and a sufficient (generic) condition for the
uniqueness of the length minimizer between two points. Hence we reduce the
optimal synthesis to solving a certain system of equations in elliptic
functions.
