Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-eight knot. The presentation is
$$\pi_1(S^3-K)= \langle a,b : yay^{-1}=b \rangle$$
Here $y=a^{-1}bab^{-1}$. Also we are given a representation into $PSL(2,\mathbb{C})$ by
$$ A =\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} \quad \mbox{and} \quad B=\begin{bmatrix} 1 & 0\\ -\omega & 1 \end{bmatrix} $$
Hhere $\omega$ is a non-trivial third root of unity. Taking $\Gamma = \langle A,B \rangle$ we find that $\Gamma$ acts on hyperbolic space $\mathbb{H}^3$ (thinking of $\mathbb{H}^3$ as the upper half space model). Now my question is the following:
Are there two elements $\alpha$ and $\beta$ in $\Gamma$ such that $\alpha$ is hyperbolic (the trace squared is real and greater than four), such that $\beta$ is strictly loxodromic (the trace squared is not in the interval $[0, \infty)$), and such that the fixed points of $\alpha$ and $\beta$ are in the same line and the geodesics passing through the fixed points do not intersect?
(Said another way: Let $g_{\alpha}$ and $g_\beta$ be the axes of $\alpha$ and $\beta$, respectively. So above we want $g_{\alpha}$ and $g_{\beta}$ to lie in a single hyperbolic plane, but not intersect.)
