  Band topology of anomalous quantum Hall insulators can be precisely addressed
by computing Chern numbers of constituent non-degenerate bands that describe
quantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chern
numbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands of
materials preserving space-inversion ($\mathcal{P}$) and time-reversal
($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, without
detailed knowledge of underlying basis? We affirmatively answer this question
by employing a non-Abelian generalization of Stokes' theorem and describe a
manifestly gauge-invariant method for computing magnitudes of quantized $SU(2)$
Berry flux (spin-Chern number) from eigenvalues of Wilson loops. The power of
this method is elucidated by performing $\mathbb{N}$-classification of \emph{ab
initio} band structures of three-dimensional, Dirac materials. Our work
outlines a unified framework for addressing first-order and higher-order
topology of insulators and semimetals, without relying on detailed symmetry
data.
