  In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of
rank two, which is the infinitesimal generator of the diffusion generated by
$n$ Brownian motions and their $\frac{n(n-1)}2$ L\'evy area processes, which is
the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$.
In order to study contraction properties of the heat kernel, we show that, as
in the cases of the Heisenberg group and the three Brownian motion model, the
restriction of the sub-Laplace acting on radial functions (see Definition
\ref{radial fun}) satisfies a positive Ricci curvature condition (more
precisely a $CD(0, \infty)$ inequality, see Theorem \ref{positive}, 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.
It can be seen a generalization of the paper \cite{Qian2}.
