= Solution
The <Killing form> is invariant under the adjoint action:
$$
\kappa(\operatorname{Ad}_gX,\operatorname{Ad}_gY)=\kappa(X,Y).
$$
Together with $F'_{\mu\nu}=\operatorname{Ad}_gF_{\mu\nu}$, this immediately gives
$$
\kappa(F'_{\mu\nu},F'^{\mu\nu})
=\kappa(F_{\mu\nu},F^{\mu\nu}),
$$
so every positive integral power $\mathcal L_1$ is gauge invariant.
The <adjoint covariant derivative> obeys
$$
(D'_\mu F'_{\nu\rho})
=g(D_\mu F_{\nu\rho})g^{-1}.
$$
This follows either by substituting the transformation laws or by applying the covariance of $D'_\mu$ to an adjoint-valued field. A second use of invariance of the Killing form gives
$$
\kappa(D'_\mu F'_{\nu\rho},D'^{\mu}F'^{\nu\rho})
=\kappa(D_\mu F_{\nu\rho},D^{\mu}F^{\nu\rho}),
$$
which proves gauge invariance of $\mathcal L_2$.
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