  We theoretically introduce and study a three-dimensional Hofstadter model
with linearly varying non-Abelian gauge potentials along all three dimensions.
The model can be interpreted as spin-orbit coupling among a trio of Hofstadter
butterfly pairs since each Cartesian surface ($xy$, $yz$, or $zx$) of the model
reduces to a two-dimensional non-Abelian Hofstadter problem. By evaluating the
commutativity among arbitrary loop operators around all axes, we derive its
genuine (necessary and sufficient) non-Abelian condition, namely, at least two
out of the three hopping phases should be neither 0 nor $\pi$. Under different
choices of gauge fields in either the Abelian or the non-Abelian regime, both
weak and strong topological insulating phases are identified in the model.

Mesoscale and Nanoscale Physics;Quantum Gases;Optics