Recent theoretical works on effective, four-band models of three-dimensional, Dirac semimetals suggest the generic planes in momentum space, orthogonal to the direction of nodal separation, and lying between two Dirac points are higher-order topological insulators, supporting gapped, edge-states. Furthermore, the second homotopy classification of four-band models shows the higher-order topological insulators support quantized, non-Abelian Berrys flux and the Dirac points are monopoles of $SO(5)$ Berrys connections. Due to the lack of suitable computational scheme, such bulk topological properties are yet to be determined from the emph{ab initio} band structures of Dirac materials. In this work, we report first, comprehensive topological classification of emph{ab initio} band structures of Na$_3$Bi, by computing Wilson loops of non-Abelian, Berrys connections for several, Kramers-degenerate bands. Our work shows the quantized, non-Abelian, Berrys flux can be used as a stable, bulk invariant for describing higher-order topology and topological phase transitions.
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