The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

Given a hyperbolic quadric of 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 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 (arXiv:1405.6888).