The fixed points in the upper half space model map to vectors on the lightcone, which span the plane that intersects with the $\mathcal{I}^3$ at $g$. The details to compute this are [here][1].

  [1]: https://math.stackexchange.com/questions/499042/how-to-convert-between-the-hyperboloid-model-and-the-poincare-patch-for-mathbb