The Berry Curvature is defined as (for invariant gauge transformations)
$$F_{ij} = [\partial_i, A_j] - [\partial_j,A_i] + [A_i,A_j]$$
The gauge covariance satisfies the transformation
$$A_i \rightarrow g^{-1}A_ig + g^{-1}\partial_ig$$
Where $$g \in U(N)$$ , in which $$N$$ is the degeneracy satisfying the transformation of the curvature
$$F_{ij} \rightarrow g^{-1}F_{ij}g$$
But this is not really pertinent to my question, but it is interesting to note that the initial part of the first equation describing the curvature will be invariant with respect to Abelian gauge transformations and not the whole $$U(N)$$ group (see reference).
The extra commutator in the first equation $$[A_i,A_j]$$ arises as a property of the gauge transformations - and in an earlier question posted at physics.stack, I asked a question concerning the similarities between the Berry curvature and the curvature tensor
Berry Curvature and Curvature Tensor
So I am almost at my question, I was informed there was actually a deep physical meaning why both the Berry curvature and the curvature share common dynamics, because they are both rooted in the same ''beautiful'' theory. This (is an oversimplification) at this moment in time to make things short. If there is an underlying deep reason they are similar, if we take a look at the Einstein equations, with a non-zero torsion, we have
$$[\partial_i \Gamma_j] - [\partial _j\Gamma_i] + [\Gamma_i,\Gamma_j]$$
Where the final commutator $$[\Gamma_i,\Gamma_j]$$ is just $$T_{ij} = \Gamma_i,\Gamma_j - \Gamma_j,\Gamma_i$$ which is a simplification of the torsion tensor in general relativity.
If the relationships are as deep as I have been led to believe rooted from the same common physics, why is the last commutator not part of (let's call it some kind of) geometric Berry torsion? The objects are so similar, again the only difference here is the name of the connections.
Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue