= Solution
Use matrix multiplication together with the <wedge product of differential forms>, suppressing wedge symbols in this calculation. Set $F=dA+A^2$ and $a=\delta A$. The <gauge field strength> varies as $\delta F=da+Aa+aA$, the <covariant exterior derivative> of $a$. The <matrix trace> has graded cyclicity, $\operatorname{Tr}(\alpha\beta)=(-1)^{pq}\operatorname{Tr}(\beta\alpha)$ for forms of degrees $p,q$. Thus the <Chern-Simons 5-form> can be expanded as
$$
Q_5=\operatorname{Tr}\left(A(dA)^2+\frac32A^3dA+\frac35A^5\right).
$$
For example, the two cross terms in $F^2A$ both become $\operatorname{Tr}(A^3dA)$; combining the remaining two terms gives the displayed coefficients. This expansion keeps track of the noncommuting products rather than treating $A$ as a scalar one-form.
Put $B=dA$. Vary the first term and use the graded <exterior derivative> to move derivatives off $a$:
$$
\begin{aligned}
\delta\operatorname{Tr}(AB^2)
&=\operatorname{Tr}(aB^2+A\,da\,B+AB\,da)\\
&=3\operatorname{Tr}(aB^2)-d\operatorname{Tr}(AaB+ABa)\\
&=3\operatorname{Tr}(aB^2)+d\operatorname{Tr}\bigl[a(BA+AB)\bigr].
\end{aligned}
$$
Here $dB=0$. For the next term use $d(A^3)=BA^2-ABA+A^2B$. Graded cyclicity then gives
$$
\begin{aligned}
\delta\operatorname{Tr}(A^3B)
&=\operatorname{Tr}(aA^2B+AaAB+A^2aB+A^3da)\\
&=2\operatorname{Tr}\bigl[a(A^2B+BA^2)\bigr]+d\operatorname{Tr}(aA^3).
\end{aligned}
$$
Finally all five contributions to $\delta\operatorname{Tr}(A^5)$ are equal under the <matrix trace>, giving $5\operatorname{Tr}(aA^4)$. Combining these results proves the <variation of the Chern-Simons 5-form>:
$$
\boxed{\delta Q_5=3\operatorname{Tr}(aF^2)+d\Theta,\qquad
\Theta=\operatorname{Tr}\left[a\left(FA+AF-\frac12A^3\right)\right].}
$$
Consequently, on an oriented five-manifold $N$,
$$
\boxed{\delta I_{\rm CS}=3\int_N\operatorname{Tr}(\delta A\wedge F\wedge F)+\int_{\partial N}\Theta.}
$$
The boundary term vanishes for a closed manifold, for variations supported in its interior, or when the tangential boundary connection is held fixed. A different boundary variational problem would require boundary conditions or a boundary action as well.
Write $A=A^\alpha T_\alpha$ in a basis of the <Lie algebra>. Independent compactly supported variations of each component yield the bulk <Euler-Lagrange equations>
$$
\boxed{\operatorname{Tr}(T_\alpha F\wedge F)=0\quad\text{for every }\alpha.}
$$
Equivalently, with $d_{\alpha\beta\gamma}=\tfrac12\operatorname{Tr}[T_\alpha\{T_\beta,T_\gamma\}]$, these are $d_{\alpha\beta\gamma}F^\beta\wedge F^\gamma=0$. The symmetry of the two curvature two-forms accounts for the anticommutator. Writing simply $F\wedge F=0$ as a full matrix equation requires an additional assumption: variations must span a matrix algebra paired nondegenerately by the <matrix trace>. For a smaller <Lie algebra>, only its trace-paired projection is constrained. If the trace is literally taken in the adjoint representation of a <semisimple Lie algebra>, its symmetric cubic invariant vanishes, and these bulk equations are identically zero. Indeed, the adjoint generators are skew with respect to the nondegenerate invariant <Killing form>; transposition in this pairing reverses the sign of the symmetrized trace of three generators. The usual nontrivial five-dimensional theory therefore needs an invariant cubic trace that does not vanish; saying that the gauge field transforms in the adjoint alone does not specify such a trace.
Under a <Yang-Mills gauge transformation>, use $A^g=g^{-1}Ag+g^{-1}dg$, so $F^g=g^{-1}Fg$. For any Lie-algebra element $T$,
$$
\operatorname{Tr}\bigl(TF^g\wedge F^g\bigr)
=\operatorname{Tr}\bigl(gTg^{-1}F\wedge F\bigr).
$$
Since $gTg^{-1}$ is still in the <Lie algebra>, the right-hand side vanishes whenever the original field satisfies every equation. \b[Every gauge transform of a bulk solution remains a bulk solution.] This conclusion follows from covariance of the field equations; it does not require the action itself to be exactly invariant under all large gauge transformations or under transformations changing prescribed boundary data.
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