of maximal $\parallel$-cliques in $\fixout$: star directions and top directions.

Let $\pclique$ be a maximal $\parallel$-clique. Take $k_1, k_2\in\pclique$ with $k_1\neq k_2$. These are proper lines that meet in some improper point $a$. In view of \[lem:bundle\] the set $\{k_1, k_2\}$ is never a maximal clique so, there must be a line in $\pclique$ distinct from $k_1, k_2$. We have two possible cases: there is a line $l\in\pclique$ that intersects $k_1, k_2$ in two distinct improper points or not. In the first case all the lines of $\pclique$ need to intersect $k_1, k_2$, and $l$ in improper points, thus they need to lie on the plane $\gen{k_1, k_2}$. In the second case all the lines of $\pclique$ go through $a$.

The condition:

-   every two distinct proper lines, sharing two distinct points with lines of a maximal $\parallel$-clique each, intersect each other or are parallel.

characterizes top directions.

\[prop:stars\] A maximal $\parallel$-clique satisfies $(\ast)$ iff it is a top direction.

Let $\pclique$ be a maximal $\parallel$-clique satisfying $(\ast)$. Suppose to the contrary that $\pclique$ is a star direction. By \[lem:bundle\] take three pairwise distinct non-coplanar lines $k_1, k_2, k_3\in\pclique$ and assume that $a\in k_1, k_2, k_3$ is the improper point 