  We present a general methodology towards the systematic characterization of
crystalline topological insulating phases with time reversal symmetry (TRS).~In
particular, taking the two-dimensional spinful hexagonal lattice as a proof of
principle we study windings of Wilson loop spectra over cuts in the Brillouin
zone that are dictated by the underlying lattice symmetries.~Our approach finds
a prominent use in elucidating and quantifying the recently proposed
``topological quantum chemistry" (TQC) concept.~Namely, we prove that the split
of an elementary band representation (EBR) by a band gap must lead to a
topological phase.~For this we first show that in addition to the Fu-Kane-Mele
$\mathbb{Z}_2$ classification, there is $C_2\mathcal{T}$-symmetry protected
$\mathbb{Z}$ classification of two-band subspaces that is obstructed by the
other crystalline symmetries, i.e.~forbidding the trivial phase. This accounts
for all nontrivial Wilson loop windings of split EBRs \textit{that are
independent of the parameterization of the flow of Wilson loops}.~Then, we show
that while Wilson loop winding of split EBRs can unwind when embedded in
higher-dimensional band space, two-band subspaces that remain separated by a
band gap from the other bands conserve their Wilson loop winding, hence
revealing that split EBRs are at least "stably trivial", i.e. necessarily
non-trivial in the non-stable (few-band) limit but possibly trivial in the
stable (many-band) limit.~This clarifies the nature of \textit{fragile}
topology that has appeared very recently.~We then argue that in the many-band
limit the stable Wilson loop winding is only determined by the Fu-Kane-Mele
$\mathbb{Z}_2$ invariant implying that further stable topological phases must
belong to the class of higher-order topological insulators.
