I have asked this question few days ago in MathStackExchange but I got only one response which gave a partial answer to my question, so I decided to ask it here.

I am reading Kulkarni's "Proper action and Pseudo-Riemannian space forms" article.

His work describes the action on the general space $S^{p,q}$, but I became interested in particular case:

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 its generators, or decribe the finitely generated subgroups of $O(2,1)$.

Thank you in advance!