Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge transformation. Denote $\partial=\nabla^{1,0}$ and $\overline{\partial}=\nabla^{0,1}$. Then
$g.\nabla=\nabla-(\overline{\partial}g)g^{-1}+((\overline{\partial}g)g^{-1})^\dagger=\nabla-(\overline{\partial}g)g^{-1}+(g^{-1}\partial g)$
since $g$ is self adjoint.
Let $F_\nabla$ be the curvature of $\nabla$. We have the formula
$F_{\nabla+A}=F_\nabla+\nabla A+\frac{1}{2}[A,A].$
How to show
$F_{g.\nabla}=F_{\nabla}-\partial((\overline{\partial}g)g^{-1})+\overline{\partial}(g^{-1}(\partial g))-(\overline{\partial}g)g^{-2}(\partial g)+g^{-1}(\partial g)(\overline{\partial}g)g^{-1}$?
This is from page 8 of Donaldson's proof of Narasimhan-Seshadri theorem.
This kind of computation is really new for me. For example, how to compute even just $[(\overline{\partial}g)g^{-1},(\overline{\partial}g)g^{-1}]$?
Thank you.