Intersection numbers of the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3)

In this paper, we study and characterise the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3). We begin by describing the possible intersections of subspaces of PG(7,q^3) with T(q^3,q). Then, we provide conditions on a set of lines L which ensures that L forms the line set of a naturally embedded twisted triality hexagon. This work follows up on similar results for the split Cayley hexagon by J. A. Thas and H. Van Maldeghem (2008) and F. Ihringer (2014).