Let $H:=\{(x_0,x_1,x_2)\in\mathbb{R}^3\mid -x_0^2+x_1^2+x_2^2=-1\}$ be the hyperbolic space with metric $g_{hip}$ induced by the Lorenz inner product $g_{Lor}=-dx_0^2+dx_1^2+dx_2^2$. Find a bijection between $H$ and $X:=\{M\in\mathbb{R}^{2\times 2}\mid M^t=M,\,\det(M)=1\}$ and show that $\text{SL}(2,\mathbb{R})$ acts on $H$ by isometries.
I've found the following correspondence: $(x_0,x_1,x_2)\in H\mapsto\left(\begin{array}{ll}(x_0+x_1) & x_2\\x_2 & (x_0-x_1)\end{array}\right)\in X$ and conversely $\left(\begin{array}{ll}a & b\\b & c\end{array}\right)\in X\mapsto\left(\frac{a+c}{2},\frac{a-c}{2},b\right)\in H$, which are mutually inverse.
I've also defined an action on $X$ given by: \begin{align*} \rho:\text{SL}(2,\mathbb{R})&\to \text{Bij}(X)\\ M&\mapsto (A\mapsto M^tAM) \end{align*} I've checked that $\rho$ is indeed a group action, but I don't know how to prove that this will be translated into an isometry of $H$.
Any suggestions?