Near hexagons and triality (Q1878962)

The authors give a very nice purely geometric characterization of the laxly embedded Moufang generalized hexagons of mixed type \(G_2\) and of regular type \(G_2\) (in projective \(6\)-space), of types \(^3D_4\) and \(^6D_4\) (in projective \(7\)-space), and of the lax embeddings of rank 3 polar spaces in the rank 4 polar space of type \(D_4\) by applying triality to a standard embedding. The assumptions are very geometric, starting with an arbitrary partial linear space \(\Pi\) laxly embedded in a nondegenerate orthogonal polar space \(\mathcal{S}\). The geometric requirements are in the style of (1) every point of \(\Pi\) is collinear in \(\mathcal{S}\) with at least one point of any line of \(\Pi\); or (2) distinct lines of \(\Pi\) in a common plane of \(\mathcal{S}\) meet in a point, etc. The axioms are chosen in such a way that they quickly lead to \(\Pi\) being a generalized hexagon or contain quadrangles, and that in case \(\Pi\) is a generalized hexagon, the embedding is regular. Then the authors appeal to a theorem of the reviewer and the second author [Can. J. Math. 56, No. 5, 1068--1093 (2004; Zbl 1072.51006)] to finish that part of the proof. The case with quadrangles requires more work and is the heart of the paper.