Consider we have a principal $G$-bundle $P$ over a closed manifold $V$. Denote $\mathfrak{g}_P$ by the associated bundle $P\times_G \mathfrak{g}$ where $G$ acts by adjoint action. Denote $\mathscr{G}$ by the gauge group.
For the following statement,
I wonder what is the standard $L^2$ metric on space of 1-forms with values in adjoint bundle. Since it seems to me we need a fiberwise metric on the adjoint bundle to define it, but I don't see the "standard choice".