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Use Euclidean coordinates $(x^1,x^2,x^3,x^4)$ and orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$. For $D_{\mu }={\partial }_{\mu }+A_{\mu }$, the gauge curvature is $F_{\mu \nu }=[D_{\mu },D_{\nu }]$. The ASDYM equations are $F=-\ast F$, equivalentlyPut $w=x^1+i{x}^2$, $z=x^3+i{x}^4$, with ${\partial }_w=({\partial }_1-i{\partial }_2)/2$ and similarly for $z$. The bars label the complex conjugate coordinates on the Euclidean real slice; after complexification they can be treated as independent coordinates. A Lax pair for the anti-self-dual Yang-Mills equations isAt infinity rescale the operators and use the other affine coordinate. Their commutator isCompatibility for every $\lambda$ is therefore precisely $F_{wz}=F_{\bar{w}\bar{z}}=0$ and $F_{w\bar{w}}+F_{z\bar{z}}=0$. To verify the real component signs,The conjugate equation supplies the remaining real components. This is compatibility of an overdetermined system, not an additional field equation on $\Psi$ for only one value of the spectral parameter.

Impose invariance under translations in $x^1,x^2,x^3$, writing $t=x^4$. In a symmetry-adapted gauge the potentials depend only on $t$. A further $t$-dependent Yang-Mills gauge transformation sets $A_4=0$ locally, by solving the usual parallel transport ordinary differential equation; it does not reintroduce spatial dependence. ThenThe three real equations above become ${\dot{A}}_1=[A_2,A_3]$, ${\dot{A}}_2=[A_3,A_1]$ and ${\dot{A}}_3=[A_1,A_2]$, namelyThese are the Nahm equations. The orientation chosen above gives the requested plus sign; reversing orientation exchanges self-duality of gauge curvature and anti-self-duality of gauge curvature. Translation invariance here is a local dimensional reduction and is not a claim of finite four-dimensional action over all three translation directions.

To prove conservation, abbreviate $P=A_1+i{A}_2$, $Q=A_1-i{A}_2$ and $R=A_3$, so $A(\lambda )=P+2R\lambda -Q{\lambda }^2$ and $B(\lambda )=-iR+iQ\lambda$. Expanding the commutator, including the cancellation of the cubic term, givesThe Nahm equations giveThus the polynomial Lax representation of the Nahm equations isTake a finite-dimensional matrix representation of the Lie algebra and its complexification. For every positive integer $p$, differentiating the product and using cyclicity of the matrix trace givesThis proves the trace invariants of a Lax equation directly. Since $\mathrm{Tr}(A(\lambda {)}^p)$ is a polynomial of degree at most $2p$ in the affine coordinate $\lambda$, its derivative vanishes identically only if the derivative of every coefficient vanishes. Therefore every trace-polynomial coefficient is independent of $t$. At projective infinity these polynomials are sections of $\mathcal{O}(2p)$; they need not be constant functions of $\lambda$ on the whole projective sphere.

On an oriented pseudo-Riemannian vector space $({\mathbb{R}}^n,\eta ,\mathrm{vol})$, the Hodge star operator is the unique linear mapsatisfyingfor all $p$-forms $\alpha ,\beta$. If $s$ is the number of negative metric directions, thenon $p$-forms under this convention. For the two-forms in the question,whereasThe 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

Write $w=x^1+i{x}^2$ and $z=x^3+i{x}^4$. The metric is conformal to the standard Euclidean metric, and the Hodge star on middle-degree differential forms is conformally invariant. Taking $e^{ij}=d{x}^i\wedge d{x}^j$, the three real forms arebecause ${\omega }_1+i{\omega }_2=dw\wedge dz$. For the orientation specified by
$dw\wedge dz\wedge d\bar{w}\wedge d\bar{z}$, one hasThe corresponding relations for the complementary basis forms immediately giveThus 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 areIntroduce the spectral parameter $\lambda$ and the linear operatorsTheir commutator isTherefore the Lax pair for the anti-self-dual Yang-Mills equationsis 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 equationshave an invertible solution $g$. Applying the associated gauge transformation setsThis conclusion is local; global topology can obstruct a single such gauge over the whole space.

Let $\zeta \in {\mathbb{CP}}^1$ be the spectral parameter. Define the two covariant differential operatorsThe auxiliary systemis compatible exactly when $[L(\zeta ),M(\zeta )]=0$ for every $\zeta$. The constant and quadratic coefficients giveand their equivalent conjugate equations, while the linear coefficient givesSince $F_{i\tau }=-F_{\tau i}$, these are preciselyThusis a Lax pair for the anti-self-dual Yang-Mills equations. In temporal gauge one simply sets $D_{\tau }={\partial }_{\tau }$.