Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$ We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where $G=\operatorname{diag}(-1,-1,1)$ leaves the pseudosphere invariant. Now we are interested in the following:
i. How can we show that $O_+(1,2)=\{ A: a_{33}>0 \}$ is a subgroup and, more importantly, maps the upper cone to the upper cone?
ii. What is the relation between the groups $O_+(1,2)$ and $\operatorname{SL}(2,\mathbb{R})$ ?