J.-P. Gauthier, Yu. L. Sachkov, “On the free Carnot (2,3,5,8) group”, Programmnye Sistemy: Teoriya i Prilozheniya, 6:2 (2015), 45–61
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 , 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.