Symmetric-matrix model of the two-sheeted hyperboloid ($B=\left(\begin{array}{cc}t+x& y\\ y& t-x\end{array}\right),\det B=t^2-x^2-y^2$)

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.