= Solution
Write $w=x^1+ix^2$, $z=x^3+ix^4$. The <Euclidean metric> is $\sum_a(dx^a)^2$, and the printed four-form is $4\,dx^1\wedge dx^2\wedge dx^3\wedge dx^4$, so it specifies the usual positive <orientation>. The normalized metric <volume form> is one quarter of that expression. The <self-dual frame in complex Euclidean coordinates> is
$$
\begin{aligned}
\omega_1&=dx^1\wedge dx^3-dx^2\wedge dx^4,\\
\omega_2&=dx^1\wedge dx^4+dx^2\wedge dx^3,\\
\omega_3&=2(dx^1\wedge dx^2+dx^3\wedge dx^4).
\end{aligned}
$$
These are real and have the required complex combinations. For the <Hodge star operator> with this <orientation>,
$$
*(dx^1\wedge dx^2)=dx^3\wedge dx^4,\qquad
*(dx^1\wedge dx^3)=-dx^2\wedge dx^4,\qquad
*(dx^1\wedge dx^4)=dx^2\wedge dx^3,
$$
and applying $*$ again gives the reverse relations. Thus $*\omega_i=\omega_i$. They are linearly independent, while the $+1$ eigenspace of $*$ on <2-forms> has dimension three, proving \b[they span $\Lambda^2_+$]. The TeX aid incorrectly reads the subscript as $1$.
The <ASDYM equations> require the self-dual projection of the <gauge field strength> to vanish. Orthogonality to $\omega_1,\omega_2$ sets its $(2,0)$ and $(0,2)$ parts to zero; orthogonality to $\omega_3$ removes the trace of its $(1,1)$ part. Explicitly, if $F=\sum_{a<b}f_{ab}\,dx^a\wedge dx^b$, these conditions are $f_{13}-f_{24}=0$, $f_{14}+f_{23}=0$ and $f_{12}+f_{34}=0$. Since
$$
F_{wz}=\tfrac14\{f_{13}-f_{24}-i(f_{14}+f_{23})\},\qquad
F_{w\bar w}+F_{z\bar z}=\tfrac i2(f_{12}+f_{34}),
$$
and the conjugate equation supplies the other complex component for a real <curvature form of a connection>, the equivalent system is
$$
\boxed{F_{wz}=0,\qquad F_{w\bar w}+F_{z\bar z}=0,\qquad F_{\bar w\bar z}=0.}
$$
To obtain the <complex potential reduction of anti-self-dual Yang-Mills>, set $D_w=\partial_w+A_w$, $D_z=\partial_z+A_z$. The first equation is the integrability condition $[D_w,D_z]=0$. Locally it allows an invertible complex matrix $s$ satisfying $\partial_ws=-A_ws$ and $\partial_zs=-A_zs$. The <Yang-Mills gauge transformation> $A\mapsto s^{-1}As+s^{-1}ds$ consequently gives $A_w=A_z=0$. This is a complex gauge; a real compact <gauge group> alone generally cannot implement it.
In this gauge the second equation becomes
$$
\partial_wA_{\bar w}+\partial_zA_{\bar z}=0.
$$
Thus the one-form $A_{\bar w}\,dz-A_{\bar z}\,dw$ is closed with respect to the exterior derivative in the $(w,z)$ directions. The local complex version of the <Poincare lemma> gives a potential $K$ such that
$$
\boxed{A_w=A_z=0,\qquad A_{\bar w}=\partial_zK,\qquad A_{\bar z}=-\partial_wK.}
$$
Because the gauge transformation is complex, $K$ is generally valued in the <complexification of a Lie algebra> $\mathfrak g_{\mathbb C}$; the printed $\mathfrak g$ must be understood in that sense. The elementary reduction is local, and the transformed fields retain a reality condition inherited from the original real connection. For the usual compact matrix gauge groups and smooth fields on all of $\mathbb R^4=\mathbb C^2$, the gauge and potential can also be chosen globally if no condition at infinity is imposed. The flat partial connection defines a holomorphic <principal bundle> on the conjugate complex space. That base is a contractible <Stein manifold>, so the <Oka-Grauert principle> gives a global trivialization and hence a global complex gauge. After that trivialization, the conjugate of <Stein vanishing for the Dolbeault cohomology of functions> gives a global primitive $K$, component by component in $\mathfrak g_{\mathbb C}$. Prescribed framing or decay at infinity requires a separate compatibility check and is not automatically preserved by this gauge.
Substitute the potential into the remaining <ASDYM equations> component:
$$
\begin{aligned}
F_{\bar w\bar z}
&=\partial_{\bar w}(-K_w)-\partial_{\bar z}K_z+[K_z,-K_w]\\
&=-K_{w\bar w}-K_{z\bar z}+[K_w,K_z].
\end{aligned}
$$
Therefore all three <ASDYM equations> reduce to the single <ASDYM potential equation>
$$
\boxed{K_{w\bar w}+K_{z\bar z}-[K_w,K_z]=0.}
$$
The sign follows directly from $A_{\bar z}=-K_w$; changing a potential convention would change the displayed commutator sign. Conversely, this equation and the displayed gauge reconstruction make all three curvature conditions vanish, subject to the inherited reality condition when a real <gauge field> is required.
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