= Solution
Use <Lie algebra> generators with $[T_b,T_c]=f^a{}_{bc}T_a$, and define the adjoint <gauge covariant derivative> by $(D_\mu X)^a=\partial_\mu X^a+f^a{}_{bc}A_\mu^bX^c$. Varying the <gauge field strength> gives $\delta F_{\mu\nu}=D_\mu\delta A_\nu-D_\nu\delta A_\mu$. Antisymmetry followed by <integration by parts> therefore gives
$$
\delta S=\frac1{g^2}\int F^{a\mu\nu}(D_\mu\delta A_\nu)^a\,d^4x=-\frac1{g^2}\int(D_\mu F^{\mu\nu})^a\delta A_\nu^a\,d^4x.
$$
For variations vanishing at the boundary, \b[the <Yang-Mills equations>] are
$$
\boxed{(D_\mu F^{\mu\nu})^a=0.}
$$
The <Jacobi identity> for the commutators of <gauge covariant derivatives>, using $[D_\mu,D_\nu]X=[F_{\mu\nu},X]$, gives \b[the <gauge-theory Bianchi identity>]:
$$
\boxed{D_\mu F_{\nu\rho}+D_\nu F_{\rho\mu}+D_\rho F_{\mu\nu}=0,\qquad D_\mu\widetilde F^{\mu\nu}=0.}
$$
In the second form of the <gauge-theory Bianchi identity> $\widetilde F^{\mu\nu}=\tfrac12\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$. The identity is a geometric consequence of the definition of curvature, not a second dynamical field equation.
For the usual non-Abelian <Yang-Mills theory>, the quantum <running coupling> becomes strong at low energies. <Dimensional transmutation> generates a scale absent from the classically scale-invariant equations. The expected confining dynamics and massive colour-singlet spectrum involve <confinement> and a <mass gap>, rather than freely propagating weakly coupled coloured waves. These nonperturbative effects are not captured by simply solving the classical equations; this is not a claim of a mathematical proof of the Yang-Mills <mass gap>.
Write the gauge condition as $\chi^a[A]=f^a[A]$ to distinguish it from the <structure constants>. Under an infinitesimal <gauge transformation>, $\delta_\omega A_\mu=D_\mu\omega$. Define the <Faddeev-Popov operator> by $\mathcal M^{ab}=\delta\chi^a[A+D\omega]/\delta\omega^b|_{\omega=0}$. In a Euclidean convention, \b[the gauge-fixing and ghost additions] can be chosen as
$$
\boxed{S_{gf}=\frac1{2\xi g^2}\int d^4x\,\chi^a\chi^a,\qquad S_{gh}=\int d^4x\,\overline c^{\,a}\mathcal M^{ab}c^b.}
$$
A nonlocal kernel would require a double integral; a local functional $\chi$ instead makes $\mathcal M$ a local differential operator. Choosing a local gauge functional thus retains a local <gauge-fixed action> and the ordinary local interaction structure of <perturbative quantum field theory>.
The fields $c^a$ and $\overline c^{\,a}$ are independent <Grassmann-valued fields> in the <Adjoint representation>, Lorentz scalars with <ghost numbers> $+1$ and $-1$. They are the <Faddeev-Popov ghost fields>: their functional integral produces $\det\mathcal M$, the <Faddeev-Popov determinant> correcting the volume element along a <gauge orbit>. They are internal fields, not physical asymptotic particles. Their statistics supply a minus sign for every closed <ghost loop>. No auxiliary field is needed in the displayed gauge-fixing representation.
In <axial gauge>, $\chi^a=n^\mu A_\mu^a$, so
$$
\mathcal M^{ac}=n\cdot\partial\,\delta^{ac}+f^a{}_{bc}\,n^\mu A_\mu^b,
\qquad S_{gh}=\int d^4x\,[\overline c^{\,a}n\cdot\partial c^a+f^a{}_{bc}\overline c^{\,a}n^\mu A_\mu^bc^c].
$$
With the <Fourier transform> convention $e^{ik\cdot x}$, \b[the ghost Feynman rules] in the original normalization of $A$ are
$$
\boxed{\langle c^a(k)\overline c^{\,b}(-k)\rangle_0=\frac{\delta^{ab}}{i\,n\cdot k},\qquad V_{\overline c^a A_\mu^b c^c}=-f^a{}_{bc}n_\mu.}
$$
Momentum conservation accompanies the vertex; each <ghost loop> has the extra minus sign. With a canonically normalized field $A=gA_{\mathrm{can}}$, the vertex is instead $-g f^a{}_{bc}n_\mu$. The <gauge fixing> term is quadratic and introduces no further interaction vertex.
The Euclidean quadratic kernel of $A$ is $g^{-2}[k^2\delta_{\mu\nu}-k_\mu k_\nu+\xi^{-1}n_\mu n_\nu]$. For $n\cdot k\ne0$, its inverse is
$$
D_{\mu\nu}^{ab}(k)=\frac{g^2\delta^{ab}}{k^2}\left[\delta_{\mu\nu}-\frac{k_\mu n_\nu+n_\mu k_\nu}{n\cdot k}+\frac{(n^2+\xi k^2)k_\mu k_\nu}{(n\cdot k)^2}\right].
$$
This follows by multiplication with the kernel: the $k^2\delta-kk$ piece gives $I-nk^T/(n\cdot k)$, and the $\xi^{-1}nn$ piece supplies the missing $nk^T/(n\cdot k)$. \b[The strict <axial-gauge propagator>] is therefore
$$
\boxed{D_{\mu\nu}^{ab}(k)\big|_{\xi=0}=\frac{g^2\delta^{ab}}{k^2}\left[\delta_{\mu\nu}-\frac{k_\mu n_\nu+n_\mu k_\nu}{n\cdot k}+\frac{n^2 k_\mu k_\nu}{(n\cdot k)^2}\right].}
$$
It obeys $n^\mu D_{\mu\nu}=0$. Every ghost attachment to an internal gauge propagator therefore vanishes in the strict limit; equivalently, $\mathcal M=n\cdot\partial$ on the gauge slice. Thus the interacting <Faddeev-Popov ghosts> decouple. Residual gauge transformations and the poles at $n\cdot k=0$ require compatible boundary conditions and an <axial-gauge pole prescription>.
After <Wick rotation>, the corresponding Minkowski propagator has $\delta_{\mu\nu}$ replaced by $\eta_{\mu\nu}$ and the usual overall factor $-i$, with a compatible causal prescription. The fixed $n$ makes the gauge-fixed propagator nonmanifestly Lorentz covariant. Nevertheless, <gauge invariance> makes physical observables independent of this gauge-choice vector, so their perturbative predictions are <Lorentz invariant>. Already at tree level, contraction with conserved external currents removes all terms containing $k_\mu$ or $k_\nu$, leaving $g^2j\cdot j'/k^2$. The quantum <Ward identities> give the analogous cancellation in complete physical amplitudes; arbitrary gauge-dependent Green functions need not be independent of $n$.
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