Sub-Riemannian structures on 3D Lie groups

We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and $A^+(\R)\times S^1$, where $A^+(\R)$ denotes the group of orientation preserving affine maps on the real line.