= Solution
On an oriented pseudo-Riemannian vector space $(\mathbb R^n,\eta,\operatorname{vol})$, the <Hodge star operator> is the unique linear map
$$
*:\Lambda^p\longrightarrow\Lambda^{n-p}
$$
satisfying
$$
\alpha\wedge *\beta=\langle\alpha,\beta\rangle_\eta\operatorname{vol}
$$
for all $p$-forms $\alpha,\beta$. If $s$ is the number of negative metric directions, then
$$
*^2=(-1)^{p(n-p)+s}
$$
on $p$-forms under this convention. For the two-forms in the question,
$$
\sigma\wedge\mu=(*\sigma)\wedge\mu
=\langle\mu,\sigma\rangle_\eta\operatorname{vol},
$$
whereas
$$
\sigma\wedge\mu=-\sigma\wedge(*\mu)
=-\langle\sigma,\mu\rangle_\eta\operatorname{vol}.
$$
The induced inner product is symmetric, so these expressions are negatives of one another. Hence a <self-dual differential form> and an <anti-self-dual differential form> are orthogonal and
$$
\boxed{\sigma\wedge\mu=0}.
$$
Write $w=x^1+ix^2$ and $z=x^3+ix^4$. The metric is conformal to the standard Euclidean metric, and the <Hodge star on middle-degree differential forms is conformally invariant>. Taking $e^{ij}=dx^i\wedge dx^j$, the three real forms are
$$
\omega_1=e^{13}-e^{24},
\qquad
\omega_2=e^{14}+e^{23},
\qquad
\omega_3=2(e^{12}+e^{34}),
$$
because $\omega_1+i\omega_2=dw\wedge dz$. For the orientation specified by
$dw\wedge dz\wedge d\bar w\wedge d\bar z$, one has
$$
*e^{13}=-e^{24},\qquad
*e^{14}=e^{23},\qquad
*e^{12}=e^{34}.
$$
The corresponding relations for the complementary basis forms immediately give
$$
\boxed{*\omega_k=\omega_k,\qquad k=1,2,3}.
$$
Thus these forms give a real basis of the self-dual two-forms.
Let $D_\alpha=\partial_\alpha+A_\alpha$ be the <gauge covariant derivative>. In these complex coordinates the <Anti-self-dual Yang-Mills equations> are
$$
F_{wz}=0,\qquad
F_{\bar w\bar z}=0,\qquad
F_{w\bar w}+F_{z\bar z}=0.
$$
Introduce the <spectral parameter> $\lambda$ and the linear operators
$$
L(\lambda)=D_w-\lambda D_{\bar z},
\qquad
M(\lambda)=D_z+\lambda D_{\bar w}.
$$
Their commutator is
$$
[L,M]
=F_{wz}
+\lambda(F_{w\bar w}+F_{z\bar z})
+\lambda^2F_{\bar w\bar z}.
$$
Therefore the <Lax pair for the anti-self-dual Yang-Mills equations>
$$
L(\lambda)\Psi=0,\qquad M(\lambda)\Psi=0
$$
is compatible for every $\lambda$ exactly when the ASDYM equations hold.
In particular, $F_{wz}=0$ says that the connection restricted to each $(w,z)$ surface is a <flat connection>. On a simply connected coordinate patch, the compatible equations
$$
(\partial_w+A_w)g=0,
\qquad
(\partial_z+A_z)g=0
$$
have an invertible solution $g$. Applying the associated <gauge transformation> sets
$$
\boxed{A_w=A_z=0}.
$$
This conclusion is local; global topology can obstruct a single such gauge over the whole space.
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