  We explicitly describe the length minimizing geodesics for a sub-Riemannian
structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses
a symmetry reduction which translates the problem into a Riemannian problem on
a two dimensional quotient space, on which projections of geodesics can be
easily visualized. As a byproduct, we obtain an alternative derivation of the
characterization of the cut-locus obtained in \cite{BoscaRossi}. We use
classification results for three dimensional right invariant sub-Riemannian
structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify
exactly automorphic structures on which our results apply.
