[proofplan]
We compute the degree-four part of the Chern-Weil total Chern form for a rank-$2$ bundle with an $SU(2)$ connection. The $SU(2)$ condition makes the curvature trace-free, so the first Chern form vanishes and the determinant expansion reduces to the trace of $F_A\wedge F_A$. The [Chern-Weil construction of Chern classes](/theorems/9769) identifies this degree-four form with the de Rham representative of $c_2(E)$, and integration over the oriented fundamental class of the closed four-manifold gives the stated second Chern number.
[/proofplan]
[step:Represent the curvature by trace-free skew-Hermitian matrices]
Let $U\subset X$ be an [open set](/page/Open%20Set) over which the $SU(2)$-bundle of frames of $E$ is trivialized, and let
\begin{align*}
F_U\in \Omega^2(U;\mathfrak{su}(2))
\end{align*}
denote the local matrix of the curvature $F_A$ in this frame. Since $\mathfrak{su}(2)$ consists of trace-free skew-Hermitian $2\times 2$ complex matrices, the local curvature matrix satisfies
\begin{align*}
\operatorname{tr}(F_U)=0.
\end{align*}
The $4$-form $\operatorname{tr}(F_U\wedge F_U)$ is independent of the chosen $SU(2)$ frame, because a change of frame conjugates $F_U$ and the ordinary matrix trace is invariant under conjugation. Hence the local forms glue to the global form
\begin{align*}
\operatorname{tr}(F_A\wedge F_A)\in \Omega^4(X;\mathbb C).
\end{align*}
[/step]
[step:Extract the degree-four part of the Chern form]
By the Chern-Weil construction of Chern classes [citetheorem:9769], with the gauge-theory convention for skew-Hermitian curvature fixed in the statement, the total Chern form of the connection is obtained locally from
\begin{align*}
c(E,A)=\det\left(I+\frac{1}{2\pi}F_U\right).
\end{align*}
Because $F_U$ has entries of even form degree, these entries commute under wedge product. Thus the usual determinant identity for a $2\times 2$ matrix applies in the graded algebra of differential forms. For every trace-free $2\times 2$ matrix $B$ with entries in a commutative algebra,
\begin{align*}
\det(B)=-\frac{1}{2}\operatorname{tr}(B^2).
\end{align*}
Applying this identity to the curvature matrix with the standard gauge-theory sign convention for skew-Hermitian curvature gives the degree-four component
\begin{align*}
c_2(E,A)=-\frac{1}{8\pi^2}\operatorname{tr}(F_U\wedge F_U).
\end{align*}
Since the right-hand side is frame-independent, this identity holds globally:
\begin{align*}
c_2(E,A)=-\frac{1}{8\pi^2}\operatorname{tr}(F_A\wedge F_A).
\end{align*}
[guided]
The determinant computation is the only place where the rank-$2$ and $SU(2)$ hypotheses enter. In a local $SU(2)$ frame, the curvature matrix $F_U$ has entries $F_{ij}\in \Omega^2(U;\mathbb C)$ and satisfies $F_{11}+F_{22}=0$. Define
\begin{align*}
\alpha:=F_{11},\qquad \beta:=F_{12},\qquad \gamma:=F_{21}.
\end{align*}
Then $F_{22}=-\alpha$, and the matrix entries of $F_U$ are determined by $\alpha,\beta,\gamma\in \Omega^2(U;\mathbb C)$.
The entries of $F_U$ are $2$-forms, so they commute with each other under wedge product: if $\eta,\theta\in \Omega^2(U;\mathbb C)$, then
\begin{align*}
\eta\wedge \theta=\theta\wedge \eta.
\end{align*}
Therefore the ordinary $2\times 2$ determinant identity is valid in the matrix algebra of differential forms, where multiplication means wedge product of form entries together with matrix multiplication. For a trace-free $2\times 2$ matrix $B$ with entries $B_{11}=a$, $B_{12}=b$, $B_{21}=c$, and $B_{22}=-a$ in a commutative algebra, one has
\begin{align*}
\det(B)=-(a\wedge a)-(b\wedge c).
\end{align*}
Matrix multiplication gives diagonal entries $a\wedge a+b\wedge c$ and $c\wedge b+a\wedge a$. Since $b$ and $c$ have even degree, $c\wedge b=b\wedge c$, and hence
\begin{align*}
\operatorname{tr}(B^2)=2(a\wedge a+b\wedge c).
\end{align*}
Combining these two identities gives
\begin{align*}
\det(B)=-\frac{1}{2}\operatorname{tr}(B^2).
\end{align*}
Now substitute the curvature matrix into the Chern-Weil determinant. Under the gauge-theory convention fixed in the theorem statement, the local total Chern form is
\begin{align*}
\det\left(I+\frac{1}{2\pi}F_U\right).
\end{align*}
Its degree-four part is the quadratic determinant term
\begin{align*}
\det\left(\frac{1}{2\pi}F_U\right)=\frac{1}{4\pi^2}\det(F_U).
\end{align*}
Using $\det(F_U)=-\frac{1}{2}\operatorname{tr}(F_U\wedge F_U)$ gives
\begin{align*}
-\frac{1}{8\pi^2}\operatorname{tr}(F_U\wedge F_U).
\end{align*}
This computation is local, but the result is global because changing an $SU(2)$ frame conjugates $F_U$, and trace is conjugation-invariant. Hence
\begin{align*}
c_2(E,A)=-\frac{1}{8\pi^2}\operatorname{tr}(F_A\wedge F_A)
\end{align*}
as a globally defined $4$-form on $X$.
[/guided]
[/step]
[step:Identify the Chern-Weil form with the topological second Chern class]
The Chern-Weil construction of Chern classes [citetheorem:9769] states that the closed form $c_2(E,A)$ represents the de Rham image of the topological second Chern class $c_2(E)$. Therefore, in $H^4_{\mathrm{dR}}(X;\mathbb C)$,
\begin{align*}
[c_2(E,A)]=c_2(E)_{\mathrm{dR}}.
\end{align*}
Using the formula from the previous step gives
\begin{align*}
c_2(E)_{\mathrm{dR}}=\left[-\frac{1}{8\pi^2}\operatorname{tr}(F_A\wedge F_A)\right].
\end{align*}
[/step]
[step:Evaluate on the oriented fundamental class]
Since $X$ is closed and oriented of dimension $4$, it has an oriented fundamental class $[X]$. Evaluation of a degree-four de Rham class on $[X]$ is the oriented integral of any closed top-degree representative over $X$. Applying this to the representative computed above yields
\begin{align*}
c_2(E)[X]=\left\langle [c_2(E,A)],[X]\right\rangle.
\end{align*}
Substituting the Chern-Weil representative gives
\begin{align*}
c_2(E)[X]=-\frac{1}{8\pi^2}\int_X^{\mathrm{or}} \operatorname{tr}(F_A\wedge F_A),
\end{align*}
where $\int_X^{\mathrm{or}}$ denotes the oriented integration map on complex-valued top-degree forms and is applied here to $\operatorname{tr}(F_A\wedge F_A)\in\Omega^4(X;\mathbb C)$.
This is the asserted formula. Since $c_2(E)\in H^4(X;\mathbb Z)$, its evaluation on the integral fundamental class is an integer, and this integer is the instanton number
\begin{align*}
k:=c_2(E)[X].
\end{align*}
[/step]
