Let $A$ be a Yang-Mills field with $A_0 = 0$ and we also have time independent scalar field $\phi$ in the adjoint representation of our gauge group with zero potential (no mass too). I have to show that $$F=*D \phi$$ satisfies equations of motion (Hodge star is taken in Euclidean space).
So EOM for Y-M fields that I'm interested in is $$D*F=0$$ Covariant derivative acts like $$D\phi = d\phi + [A,\phi]$$ Two Hodge stars are proportional to identity so my equation is $$DD\phi=0=dd\phi + d[A,\phi]+[A,d\phi +[A,\phi]]=0+[dA,\phi]+2[A, d\phi] + [A,[A,\phi]] $$ I don't really see how that can be zero. Any tips?
Edit: Also: is $DF = 0$?