Let $\mathcal{I}^3\subset\mathbb{R}^4$ be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$ action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathcal{I}^3$. Given a matrix $\gamma=\begin{pmatrix} a & b\\ c & d \end{pmatrix}\in\mathrm{PSL}(2,\mathbb{C})$, there is a unique geodesic $g\in\mathcal{I}^3$ preserved by the action of $\gamma$. Particularly, $\gamma$ translates along and rotates around $g$ (and these quantities can be written in terms of the trace $a+d$).

I want to know how to determine the axis $g$ in terms of the entries of $\gamma$. One way to go about this would be to use the fact that $g$ is the intersection $P\cap\mathcal{I}^3$ where $P$ is a uniquely determiend Euclidean plane passing through the origin in $\mathbb{R}^4$. Perhaps we can write down two vectors in terms of $a,b,c,d$ that span this plane, using the fact that the extended action of $\gamma$ to $\mathbb{R}^4$ preserves the plane.

By comparison, in the upper half-space model $\mathcal{H}^3$, the fixed godesic will be the Euclidean half-circle or half-line orthogonal to $\partial\mathcal{H}^3$ that connects the points $\dfrac{a-d\pm\sqrt{(a-d)^2+4bc}}{2c}$, where we interpret these as $0$ and $\infty$ when $c=0$. But in the hyperboloid model, points on the boundary are not so accessible. So how do we describe the fixed geodesic there?