I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane.
I've just been told that $SO(2,1)$, the indefinite special orthogonal group, is also isomorphic to the full isometry group of the hyperbolic plane.
How can I prove this algebraically? That is, how do I find an explicit isomorphism?
I've checked a couple obvious invariants. They're both 3-dimensional, and they are both centerless, so the argument has some credibility. I just can't seem to find an explicit isomorphism.
Send $\left(\begin{array}{cc}a&b\\ c&d\end{array}\right)\in PSL(2,{\mathbb R})$ to $$\left(\begin{array}{ccc}\frac{a^2+b^2+c^2+d^2}{2}&ab+cd&\frac{a^2-b^2+c^2-d^2}{2}\\ ac+bd&ad+bc&ac-bd\\ \frac{a^2+b^2-c^2-d^2}{2}&ab-cd&\frac{a^2-b^2-c^2+d^2}{2}\end{array}\right)\in SO(2,1)$$