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 $y=a^{-1}bab^{-1}$ ($K$ the figure-eight knot) and also the representation in $PSL(2,\mathbb{C})$ is given by the matrices 

$	
A =\begin{bmatrix} 
	1 & 1\\
	0 & 1 
\end{bmatrix}
$

$
B=\begin{bmatrix} 
	1 & 0\\
-\omega  & 1
\end{bmatrix}
$

where $\omega ^3=1$ and $\Gamma = \langle A,B \rangle$, and $\Gamma$ act on $\mathbb{H}^3$ by isometry (here $\mathbb{H}^3$ is the upper half space model). Now my question is the following: 

Are there 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 do not intersect?

(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.)
 NB="lie in the same plane" I mean that it is a hyperbolic plane in upper half-space model 

Thanks in advance