New parameters for Fuchsian groups of genus $2$

We give a new real-analytic embedding of the Teichmüller space of closed Riemann surfaces of genus 2 into ${\mathbb R^6}$. The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in $PSL(2,{\mathbb R})$, where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to define the image of the embedding; the four matrices corresponding to a point in this image generate a fuchsian group representing a closed Riemann surface of genus $2$.