  We give the complete classification of left-invariant sub-Riemannian
structures on three dimensional Lie groups in terms of the basic differential
invariants. This classifications recovers other known classification results in
the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in
terms of curvature invariants of a canonical connection. Moreover, we
explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups
$SL(2)$ and $A^{+}(\mathbb{R})\times S^1$, where $A^+(\mathbb{R})$ denotes the
group of orientation preserving affine maps on the real line.
