 Solution

Identify $x=(x^0,x^1,x^2)$ with the real symmetric matrixFor $A\in SL(2,\mathbb{R})$, define $X\mapsto AX{A}^T$. This linear action preserves $\det X$, so it defines $\rho (A)\in S{O}^{+}(1,2)$, and $\rho (AB)=\rho (A)\rho (B)$. Its kernel is $\{\pm I\}$ and it is onto, hence it is a two-to-one covering homomorphism rather than an isomorphism. Consequently $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I\}\cong S{O}^{+}(1,2)$.