'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

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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 18₂ - 12₃ and 2₄14₂ - 4₃6₄ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V₂₂(37; 0, 12, 15, 10) and V₄(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.