Opposition in triality

In this article we consider a sub-geometry G of the D4 building geometry whose flags of type {1, 3, 4} are exactly those which are opposite to their image under a triality on D4, while the lines of G are certain so-called skew lines (see Definition 3.4). We prove that this rank four geometry G admits the group G2 as a flag-transitive group of automorphisms. Moreover, if the underlying field contains at least three elements, the geometry G is simply connected. Accordingly, we obtain an amalgam presentation of G2 via the rank one and two parabolics of the action of G2 on G .