A 51-dimensional embedding of the Ree–Tits generalized octagon

Abstract

We construct an embedding of the Ree–Tits generalized octagon defined over a field K in a 51-dimensional projective space over K arising from a 52-dimensional Lie algebra J of type ${\mathsf{F}}_4$ . This construction derives from a quadratic map (related to a ‘standard’ duality of ${\mathsf{F}}_4$) from the 26-dimensional module (see K. Coolsaet, Adv Geometry, to appear) into J. (This embedding is full if and only if K is a perfect field.) We provide explicit formulas for the coordinates of the points of the octagon in this embedding, in terms of their Van Maldeghem coordinates. We apply these results to compute the dimensions of subspaces generated by various special subsets of points of the octagon: the sets of points at a fixed distance from a given point or a given line and the Suzuki suboctagons. The results depend on whether K is the field of 2 elements, or not.