  As it is well known, split Cayley hexagons of order two live in the
three-qubit symplectic polar space in two non-isomorphic embeddings, called
classical and skew. Although neither of the two embeddings yields
observable-based contextual configurations of their own, {\it
classically}-embedded copies are found to fully rule contextuality properties
of the most prominent three-qubit contextual configurations in the following
sense: each set of unsatisfiable contexts of such a contextual configuration is
isomorphic to the set of lines that certain classically-embedded hexagon shares
with this particular configuration. In particular, for a doily this shared set
comprises three pairwise disjoint lines belonging to a grid of the doily, for
an elliptic quadric the corresponding set features nine mutually disjoint lines
forming a (Desarguesian) spread on the quadric, for a hyperbolic quadric the
set entails 21 lines that are in bijection with the edges of the Heawood graph
and, finally, for the configuration that consists of all the 315 contexts of
the space its 63 unsatisfiable ones cover an entire hexagon. A particular
illustration of this encoding is provided by the {\it line-complement} of a
skew-embedded hexagon; its 24 unsatisfiable contexts correspond exactly to
those 24 lines in which a particular classical copy of the hexagon differs from
the considered skew-embedded one. In connection with the last-mentioned case we
also conducted some experimental tests on a Noisy Intermediate Scale Quantum
(NISQ) computer to validate our theoretical findings.
