  In this note, we look at some hypoelliptic operators arising from nilpotent
rank 2 Lie algebras. In particular, we concentrate on the diffusion generated
by three Brownian motions and their three L\'evy areas, which is the simplest
extension of the Laplacian on the Heisenberg group $\mathbb{H}$. In order to
study contraction properties of the heat kernel, we show that, as in the case
of the Heisenberg group, the restriction of the sub-Laplace operator acting on
radial functions (which are defined in some precise way in the core of the
paper) satisfies a non negative Ricci curvature condition (more precisely a
$CD(0, \infty)$ inequality), whereas the operator itself does not satisfy any
$CD(r,\infty)$ inequality. From this we may deduce some useful, sharp gradient
bounds for the associated heat kernel.
