A proof for physicists of Proposition 5.5 in "Foundations of Differential Geometry Vol 1" by Kobayashi and Nomizu

Question

Please consider the proposition

My questions are : 

1.  How to give a proof for physicists of this proposition?

2.  What could be a physical application of this proposition?

Answer 1

A possible application of the the proposition 5.5  in physics occurs in the case of the Kapustin-Witten equation:

In this case  $\phi$ is a tensorial 1-form of type  $ad G$ and $d_{A} \phi$  is given by the proposition 5.5.

It is conjectured that the coefficients of the Jones polynomial of a knot can be computed by counting solutions of the KW equations on a half-space in $R^4$ with the generalized Nahm pole boundary conditions. 

The Jones polynomial is a Laurent series $J \left( q \right) =\sum _ na_{{n}}{q}^{n} $,  and the conjecture is that $a_ n$  is an algebraic count of the number of solutions of the KW equations with second Chern class   equal to $n$.

Reference :  https://arxiv.org/pdf/1712.00835.pdf

Answer 2

Other application of the proposition 5.5  is given in the following lemma

with 

where: