• 1 University of Pretoria [South Africa]

We classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first twostep nilpotent Lie group beyond the three-and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).

• 17B30 - Solvable, nilpotent (super)algebras
• 22E25 - Nilpotent and solvable Lie groups
• 53C17 - Sub-Riemannian geometry