New properties of length-extremals in free step-2 rank-4 Carnot groups

In the free, step-2, rank-4 sub-Riemannian Carnot group, we give a clean expression for length-extremals, we provide an explicit equation for conjugate points, we relate it with the conjectured cut-locus of the origin. Finally, we give some upper estimates for the cut-time of extremals.