On the free Carnot (2,3,5,8) group

We consider the free nilpotent Lie algebra $L$ with 2 generators, of step 4, and the corresponding connected simply connected Lie group $G$, with the aim to study the left-invariant sub-Riemannian structure on $G$ defined by the generators of $L$ as an orthonormal frame.
We compute two vector field models of $L$ by polynomial vector fields in $\mathbb{R}^8$, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in $G$ and the right-invariant frame on $G$.