Incidence matrices for the class ${\mathcal{O}}_6$ of lines external to the twisted cubic in $\mathrm{P}\mathrm{G}(3,q)$

We consider the structures of the plane-line and point-line incidence matrices of the projective space $\mathrm{P}\mathrm{G}(3,q)$ connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all $q$, even and odd, we determine the incidence matrices connected with a family of orbits of the class named ${\mathcal{O}}_6$. This class contains lines external to the twisted cubic. The considered family include an essential part of all ${\mathcal{O}}_6$ orbits, whose complete classification is an open problem.