  We present a non-abelian cousin of the model presented in [1] which induces
cosmological anisotropies on top of standard FLRW geometry. This is in some
sense doing a cosmological mean field approximation, where the mean field
cosmological model under consideration would be the standard FLRW, and the
induced anisotropies are small perturbative corrections on top of it. Here we
mostly focus on the non-abelian $SU(2)$ gauge fields coupled to the gravity to
generate the anisotropies, which can be a viable model for the axion-like
particle (ALP) dark sector. The induced anisotropies are consequences of the
non-trivial back-reaction of the gauge fields on the gravity sector, and by a
clever choice of the parametrization, one can generate the Bianchi model we
have studied in this note. We also show that the anisotropies influence the
Sachs-Wolfe effect and we discuss the implications.
