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As it is well known, split Cayley hexagons of order two live in thethree-qubit symplectic polar space in two non-isomorphic embeddings, calledclassical and skew. Although neither of the two embeddings yieldsobservable-based contextual configurations of their own, {\itclassically}-embedded copies are found to fully rule contextuality propertiesof the most prominent three-qubit contextual configurations in the followingsense: each set of unsatisfiable contexts of such a contextual configuration isisomorphic to the set of lines that certain classically-embedded hexagon shareswith this particular configuration. In particular, for a doily this shared setcomprises three pairwise disjoint lines belonging to a grid of the doily, foran elliptic quadric the corresponding set features nine mutually disjoint linesforming a (Desarguesian) spread on the quadric, for a hyperbolic quadric theset entails 21 lines that are in bijection with the edges of the Heawood graphand, finally, for the configuration that consists of all the 315 contexts ofthe space its 63 unsatisfiable ones cover an entire hexagon. A particularillustration of this encoding is provided by the {\it line-complement} of askew-embedded hexagon; its 24 unsatisfiable contexts correspond exactly tothose 24 lines in which a particular classical copy of the hexagon differs fromthe considered skew-embedded one. In connection with the last-mentioned case wealso conducted some experimental tests on a Noisy Intermediate Scale Quantum(NISQ) computer to validate our theoretical findings.

Classically-embedded split Cayley hexagons rule three-qubit contextuality with three-element contexts