= Solution
Take a local <gauge transformation> $g(x)$ acting on the adjoint field by $\Phi'=g\Phi g^{-1}$. The connection convention $D_\mu=\partial_\mu+[A_\mu,\cdot]$ is compatible with
$$
\boxed{A_\mu'=gA_\mu g^{-1}-(\partial_\mu g)g^{-1},\qquad
D_\mu'\Phi'=g(D_\mu\Phi)g^{-1}.}
$$
Indeed, differentiating $\Phi'$ gives the extra <commutator> $[(\partial_\mu g)g^{-1},\Phi']$, exactly cancelled by the inhomogeneous term in the transformed <gauge potential>.
For a direct proof of the <Yang-Mills gauge transformation> of curvature, put $\Omega_\mu=(\partial_\mu g)g^{-1}$ and $B_\mu=gA_\mu g^{-1}$. Differentiation gives
$$
\partial_\mu B_\nu=g(\partial_\mu A_\nu)g^{-1}+[\Omega_\mu,B_\nu],
\qquad
\partial_\mu\Omega_\nu-\partial_\nu\Omega_\mu=[\Omega_\mu,\Omega_\nu].
$$
Substituting $A_\mu'=B_\mu-\Omega_\mu$ into the <Yang-Mills field strength> cancels all terms involving $\Omega$ and leaves
$$
\boxed{F_{\mu\nu}'=gF_{\mu\nu}g^{-1}.}
$$
This direct calculation also applies when the adjoint action has a kernel.
Expand $D_\mu F_{\nu\rho}+D_\nu F_{\rho\mu}+D_\rho F_{\mu\nu}$. The mixed second derivatives of $A$ cancel in pairs. The first-derivative terms from differentiating $[A_\nu,A_\rho]$ cancel the terms from $[A_\mu,\partial_\nu A_\rho-\partial_\rho A_\nu]$. The remaining sum is
$$
[A_\mu,[A_\nu,A_\rho]]+[A_\nu,[A_\rho,A_\mu]]
+[A_\rho,[A_\mu,A_\nu]]=0
$$
by the <Jacobi identity>. Thus the <gauge-theory Bianchi identity> is
$$
\boxed{D_\mu F_{\nu\rho}+D_\nu F_{\rho\mu}+D_\rho F_{\mu\nu}=0.}
$$
For the variation of the <Yang-Mills action>, write $a_\mu=\delta A_\mu$. Differentiating the curvature gives $\delta F_{\mu\nu}=D_\mu a_\nu-D_\nu a_\mu$. Invariance of the <Killing form> implies
$$
\partial_\mu\kappa(X,Y)=\kappa(D_\mu X,Y)+\kappa(X,D_\mu Y).
$$
This is <invariant integration by parts for a gauge covariant derivative>. Using symmetry of the <Killing form>, antisymmetry of $F$, and compactly supported variations,
$$
\begin{aligned}
\delta S_{\mathrm{YM}}
&=\frac1{2e^2}\int d^4x\,\kappa(\delta F_{\mu\nu},F^{\mu\nu})\\
&=\frac1{e^2}\int d^4x\,\kappa(D_\mu a_\nu,F^{\mu\nu})\\
&=-\frac1{e^2}\int d^4x\,\kappa(a_\nu,D_\mu F^{\mu\nu}).
\end{aligned}
$$
Since the <Killing form> of a <semisimple Lie algebra> is nondegenerate and the variations are arbitrary, the <Yang-Mills equations> are
$$
\boxed{D_\mu F^{\mu\nu}=0.}
$$
For the constant <Yang-Mills theta term>, symmetry under exchanging the two antisymmetric index pairs gives
$$
\begin{aligned}
\delta S_\theta
&=2\theta\int d^4x\,\epsilon^{\mu\nu\rho\sigma}
\kappa(\delta F_{\mu\nu},F_{\rho\sigma})\\
&=4\theta\int d^4x\,\epsilon^{\mu\nu\rho\sigma}
\kappa(D_\mu a_\nu,F_{\rho\sigma})\\
&=-4\theta\int d^4x\,\epsilon^{\mu\nu\rho\sigma}
\kappa(a_\nu,D_\mu F_{\rho\sigma}),
\end{aligned}
$$
up to the <boundary variation of the Yang-Mills theta term>. Contracting the <gauge-theory Bianchi identity> with $\epsilon^{\mu\nu\rho\sigma}$ makes the last integrand vanish. Therefore \b[a constant theta term does not alter the bulk <Yang-Mills equations>: $D_\mu F^{\mu\nu}=0$.] Its variation is a boundary term, so this conclusion uses compact support or boundary conditions that remove that term.
Finally apply $D^\rho$ to the <gauge-theory Bianchi identity>, obtaining
$$
D^\rho D_\rho F_{\mu\nu}
=-D^\rho D_\mu F_{\nu\rho}-D^\rho D_\nu F_{\rho\mu}.
$$
The <adjoint covariant derivative> obeys $[D_\alpha,D_\beta]X=[F_{\alpha\beta},X]$. Commute $D^\rho$ past the other derivative. The differentiated divergences vanish by the <Yang-Mills equations>, leaving
$$
\begin{aligned}
D^\rho D_\rho F_{\mu\nu}
&=-[F^\rho{}_\mu,F_{\nu\rho}]
-[F^\rho{}_\nu,F_{\rho\mu}]\\
&=[F_\mu{}^\rho,F_{\nu\rho}]
+[F_{\mu\rho},F_\nu{}^\rho].
\end{aligned}
$$
In the last line the second <commutator> needs its sign tracked carefully: directly,
$$
-[F^\rho{}_\nu,F_{\rho\mu}]
=-[F_\nu{}^\rho,F_{\mu\rho}]
=[F_{\mu\rho},F_\nu{}^\rho]
=[F_\mu{}^\rho,F_{\nu\rho}].
$$
Thus the simplified sum gives the <covariant wave equation for the Yang-Mills field strength>
$$
\boxed{D^\rho D_\rho F_{\mu\nu}
=2[F_\mu{}^\rho,F_{\nu\rho}].}
$$
All spacetime derivatives and index manipulations here use the fixed flat metric.
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