= Solution
Choose orientation $\epsilon_{1234}=1$ and the anti-Hermitian basis in the question. The <Pauli matrix commutator identity> gives $[e_a,e_b]=\epsilon_{abc}e_c$. Define the <gauge curvature> and <adjoint covariant derivative> by
$$
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu],
\qquad D_\mu X=\partial_\mu X+[A_\mu,X].
$$
For a compactly supported variation $a_\mu=\delta A_\mu$,
$$
\delta F_{\mu\nu}=D_\mu a_\nu-D_\nu a_\mu.
$$
Antisymmetry and the invariant Lie-algebra inner product therefore give
$$
\delta V=4\int_{\mathbb R^4}\langle F_{\mu\nu},D_\mu a_\nu\rangle
=-4\int_{\mathbb R^4}\langle D_\mu F_{\mu\nu},a_\nu\rangle.
$$
Covariant integration by parts is valid because
$\langle X,[A,Y]\rangle=-\langle[A,X],Y\rangle$. Thus the <Euler-Lagrange field equations> are
$$
\boxed{D_\mu F_{\mu\nu}=0,\qquad
\partial_\mu F_{\mu\nu}^a+\epsilon_{abc}A_\mu^bF_{\mu\nu}^c=0.}
$$
The Euclidean <Hodge star operator> on two-forms is
$$
(*F)_{\mu\nu}=\frac12\epsilon_{\mu\nu\kappa\lambda}F_{\kappa\lambda},
\qquad *^2=1.
$$
The <Anti-self-dual Yang-Mills equations> mean $\boxed{*F=-F}$.
The <gauge-theory Bianchi identity> is
$$
\boxed{D_\mu F_{\nu\rho}+D_\nu F_{\rho\mu}+D_\rho F_{\mu\nu}=0.}
$$
To prove it, act on the defining representation with $\nabla_\mu=\partial_\mu+A_\mu$. Its commutator is multiplication by $F_{\mu\nu}$. The operator <Jacobi identity> for $\nabla_\mu,\nabla_\nu,\nabla_\rho$ becomes multiplication by the displayed cyclic sum, which must vanish. Equivalently, substitution of $F=dA+A\wedge A$ into $D_AF$ cancels the ordinary derivatives, leaving the Lie-algebra <Jacobi identity>. Contracting the cyclic identity with the alternating tensor gives $D_\mu(*F)_{\mu\nu}=0$. If $*F=-F$, this is exactly $D_\mu F_{\mu\nu}=0$, proving the first-order-to-second-order implication without assuming the field equations.
Let
$$
J=\int_{\mathbb R^4}\epsilon_{\mu\nu\kappa\lambda}
F_{\mu\nu}^aF_{\kappa\lambda}^a\,d^4x
=2\int_{\mathbb R^4}F_{\mu\nu}^a(*F)_{\mu\nu}^a\,d^4x.
$$
All squared component sums below include both orders of each antisymmetric pair, as does the printed energy. Since the <Hodge star> preserves this norm,
$$
\boxed{V=\frac12\int_{\mathbb R^4}|F+*F|^2\,d^4x-\frac J2
=\frac12\int_{\mathbb R^4}|F-*F|^2\,d^4x+\frac J2.}
$$
Consequently $V\ge |J|/2$. An anti-self-dual field has $J=-2V\le0$ and saturates the first bound. Among all finite-energy potentials with the same value of $J$, it therefore has the absolute minimum $-J/2$. For positive $J$, the saturating sign is self-duality instead; there cannot be a nonzero anti-self-dual solution in that sector. This is the <Yang-Mills instanton Bogomolny bound> in the paper's normalization.
For the topological meaning, regard $F=\tfrac12F_{\mu\nu}^ae_a\,dx^\mu\wedge dx^\nu$ as a matrix-valued two-form. Since $\operatorname{tr}(e_ae_b)=-\delta_{ab}/2$,
$$
\operatorname{tr}(F\wedge F)
=-\frac18\epsilon_{\mu\nu\kappa\lambda}
F_{\mu\nu}^aF_{\kappa\lambda}^a\,d^4x.
$$
In the <Second Chern number> convention used here,
$$
\boxed{k=\frac1{8\pi^2}\int_{\mathbb R^4}
\operatorname{tr}(F\wedge F)=-\frac{J}{64\pi^2}.}
$$
With the standard instanton boundary condition, the connection approaches a pure gauge at infinity and extends to a bundle over the compactified four-sphere. This makes $k$ an integer, the <instanton number>; reversing the orientation or using the opposite charge convention reverses its sign. The square-completion argument itself does not require quantization or a chosen framing.
One can see the boundary interpretation directly. The <Chern-Simons 3-form> obeys
$$
d\,\operatorname{tr}\left(A\wedge dA+\frac23A\wedge A\wedge A\right)
=\operatorname{tr}(F\wedge F).
$$
For an asymptotic potential $A=-dg\,g^{-1}$, the flatness identity $dA=-A\wedge A$ gives
$$
k=\frac1{24\pi^2}\int_{S^3_\infty}
\operatorname{tr}\left[(dg\,g^{-1})^{\wedge3}\right].
$$
This is the winding number of $g:S^3_\infty\to SU(2)\cong S^3$, with the corresponding orientation convention. Smooth variations preserving that boundary class cannot change it. Thus the integral in the question labels the topological sector of the <Yang-Mills instanton>, and in an anti-self-dual sector the energy is $\boxed{V=32\pi^2k}$ with $k\ge0$ in our convention.
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