The index of sub-laplacians: beyond contact manifolds

In this paper we study the following question: do sub-Laplacian type operators have non-trivial index theory on Carnot manifolds in higher degree of nilpotency? The problem relates to characterizing the structure of the space of hypoelliptic sub-Laplacian type operators, and results going back to Rothschild-Stein and Helffer-Nourrigat. In two degrees of nilpotency, there is a rich index theory by work of van Erp-Baum on contact manifolds, that was later extended to polycontact manifolds by Goffeng-Kuzmin. We provide a plethora of examples in higher degree of nilpotency where the index theory is trivial.