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In this note, we concentrate on the sub-Laplace on the nilpotent Lie group ofrank 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 isthe 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, asin the cases of the Heisenberg group and the three Brownian motion model, therestriction of the sub-Laplace acting on radial functions (see Definition\ref{radial fun}) satisfies a positive Ricci curvature condition (moreprecisely a $CD(0, \infty)$ inequality, see Theorem \ref{positive}, whereas theoperator itself does not satisfy any $CD(r,\infty)$ inequality. From this wemay deduce some useful, sharp gradient bounds for the associated heat kernel.It can be seen a generalization of the paper \cite{Qian2}.

Positive curvature property for sub-Laplace on nilpotent Lie group of rank two