Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise gives us a presentation of the fundamental group of figure-eight knot complements in $S^3$ the presentation is as follows
$\pi_1(S^3-K)= \langle a,b : yay^{-1}=b\rangle $ where $y=a^{-1}bab^{-1}$ and also the representation in $PSL(2,\mathbb C)$ is given by the metrics
$ A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$
$B= \begin{bmatrix} 1 & 1 \\ -\omega & 0 \end{bmatrix}$
and $\Gamma = \langle A,B\rangle$, and $\Gamma$ act on $\mathbb{H}^3$ by isometry (Here $\mathbb{H}^3$ upper half space model ) Now my question is, does there exist two elements in $\Gamma$ such that one of them is hyperbolic say $\alpha$ ( $trace^2$ is real and $>4$) and another on loxodromic say $\beta$ ($trace^2$ not in the interval $[0, \infty)$ )but not hyperbolic such that fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points are not intersecting each other? (More elaborately, I can say geodesics $g_{\alpha}$ passing through the fixed points of $\alpha$ and $g_{\beta}$ the geodesic passing through the fixed points of $\beta$, $g_{\alpha}$ and $g_{\beta}$ lie in the same plane, but they do not intersect.)
Thanks in advance