‘Magic’ configurations of three-qubit observables and geometric hyperplanes of the smallest split Cayley hexagon

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen–Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_2-12_3$ and $2_414_2-4_36_4$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types $\mathcal V_{22}(37;0,12,15,10)$ and $\mathcal V_4(49;0,0,21,28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773–797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.