Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the case where $n = 2$, one obtains the familiar fundamental domain for the action of $\operatorname{SL}_2(\mathbb{Z})$ on the complex upper half-plane. The case where $n = 3$ is studied in detail in the paper entitled "Hecke Operators and the Fundamental Domain for $\operatorname{SL}(3, \mathbb{Z})$" by Daniel Gordon et al.

Are there analogous computations in the literature of explicit fundamental domains for the action of the orthogonal group $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$, at least for some small values of $n$? I am particularly interested in the case where $n = 2$.

What I know: I understand that computing such fundamental domains is difficult in general. I'm aware of the construction of Borel and Harish-Chandra via Siegel domains, but I'm not sure whether it's possible to make their construction explicit in the way that Minkowski was able to do.