Let $H^n=\{X=(x_1,x_2....x_n) \in\mathbb{R}^{n,1}: <X,X>=-1, x_1>0\}$ denote the upper hyperboloid in the hyperboloid model of the hyperbolic space. The components of $O(n,1)$ that preserves $H^n$ is denoted by $PO(n,1)$.
This question is motivated from finding the conjugacy classes of $PSL_2(\mathbb{R})$ which is the isometry group of the upper half plane model of the hyperbolic space. In this case, we expect $PO(n,1)$ to be the isometry group of $H^n$.
This is nontrivial, but known. Look at theorems 1.1 and 1.2 in [this paper by Gongopadhyay and Kulkarni.][1]
Gongopadhyay, Krishnendu; Kulkarni, Ravi S., (z)-classes of isometries of the hyperbolic space, Conform. Geom. Dyn. 13, 91-109 (2009). ZBL1206.51017. [1]: https://www.ams.org/journals/ecgd/2009-13-04/S1088-4173-09-00190-8/S1088-4173-09-00190-8.pdf