 Symmetric-matrix model of the two-sheeted hyperboloid

The displayed coordinates identify determinant-one real symmetric matrices of size two with the two-sheeted hyperboloid. The special linear congruence action on symmetric matrices is transitive on each sheet, with stabilizer $SO(2)$. In particular the positive sheet is the homogeneous space $SL(2,\mathbb{R})/SO(2)$, a model of the hyperbolic plane.