  In this note I present some properties of sub-Laplaceans associated with a
collection of smooth vector fields satisfying H\"ormander's finite rank
assumption. One notable aspect of the paper is the development of the
fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an
extension problem inspired to the famous 2007 work of Caffarelli and Silvestre
for the standard Laplacean. A key tool is an extension problem for the
fractional heat equation for which I compute the relevant Poisson kernel. I
then use the latter to: 1) find the Poisson kernel for the time-independent
case; and 2) solve the extension problem.
