Simplify Yang-Mills Equation of Motion in the 1-form gauge field $A$

Question

We know the Yang-Mills theory Equation of Motion (eom) without source

$$* D * F = * (d (* F ) + [A, (* F )])= 0.$$

My question is that what are the most simple form we can boil down this eom to its minimal?

$$* (d (* (d A + A \wedge A) ) + [A, (* (d A + A \wedge A) )])$$

$$=* d * d A + * d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A=0$$

This is what I get. How can we massage it further in order to make it as simple as possible but similar to the Maxwell's ---

$$* d * d A + ... =0?$$ What is the simplest form of $...$ term?

p.s. What I got so far is that $...$ term is $$C=* d * (A \wedge A) +A( * (d a + A \wedge A) ) - (* (dA + A \wedge A) ) A.$$ Do we have better way to simplify this complicated $C$ in terms of $A$?

This post imported from StackExchange Physics at 2020-11-09 19:27 (UTC), posted by SE-user annie marie heart

Comments

$\star d \star dA$ is not gauge invariant, so why do you want to isolate it?

This post imported from StackExchange Physics at 2020-11-09 19:27 (UTC), posted by SE-user Ryan Thorngren

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$F$ is also not gauge invariant, but only covariant. I want to see $C just for comparing the higher order terms.

This post imported from StackExchange Physics at 2020-11-09 19:27 (UTC), posted by SE-user annie marie heart

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$F$ transforms homogeneously so it's much easier to understand than $\star d \star A$.

This post imported from StackExchange Physics at 2020-11-09 19:27 (UTC), posted by SE-user Ryan Thorngren