= Solution
Identify $x=(x^0,x^1,x^2)$ with the real symmetric matrix
$$
X=\phi(x)=\begin{pmatrix}x^0+x^1&x^2\\x^2&x^0-x^1\end{pmatrix},
\qquad \det X=(x^0)^2-(x^1)^2-(x^2)^2.
$$
For $A\in SL(2,\mathbb R)$, define $X\mapsto AXA^T$. This linear action preserves $\det X$, so it defines $\rho(A)\in SO^+(1,2)$, 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 $PSL(2,\mathbb R)=SL(2,\mathbb R)/\{\pm I\}\cong SO^+(1,2)$.
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