I am working with one sheeted hyperboloid $S^{1,1}$ in $\mathbb{R}^3$ with equation $$x^2+y^2-z^2=1.$$ $O(2,1)$ is defined to be a group of $Q$-orthogonal transformations preserving $S^{1,1}$.
$Q$-orthogonality means preserving of the bilinear form $$b[(x_1,y_1,z_1), (x_2,y_2,z_2)] = x_1x_2+y_1y_2-z_1z_2.$$
$\textbf{My question is}$: are there any references/articles which describe the general form of elements of $O(2,1)$, or decribe the finitely generated subgroups of $O(2,1)$.
Thank you in advance!
https://www.physicsforums.com/threads/why-is-lorentz-group-in-3d-sl-2-r.764072/ explains that $SO(2,1) \equiv SL(2,\mathbb{R})$, and explains the conversion between them. Since you also allow non-proper transformations (those with a z inversion), an element of $O(2,1)$ is in general one of the above $SL(2,\mathbb{R})$ composed with an optional z reflection.