The complement of binary Klein quadric as a combinatorial Grassmannian (Q2352959)

Summary: Given a hyperbolic quadric of \(\mathrm{PG}(5,2)\), there are 28 points off this quadric and 56 lines skew to it. It is shown that the \((28_6,56_3)\)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type \(G_2(8)\). It is also pointed out that a set of seven points of \(G_2(8)\) whose labels share a mark corresponds to a Conwell heptad of \(\mathrm{PG}(5,2)\). Gradual removal of Conwell heptads from the \((28_6,56_3)\)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras \textit{M. Saniga} et al. [``Cayley-Dickson algebras and finite geometry'', Preprint, \url{arXiv:1405.6888}].