A doubt about Yang-Mills equation in trivial $U(1)$-bundles

I know that the Yang-Mills equation is $delta F=0$ where $delta$ is the adjoint operator of the covariant derivative and $F$ is the curvature of a principal connection. In a trivial $U(1)$-bundle (for example $mathbb{R}^4times U(1)$ as the one used in electrodynamics), $F=dA$ for some 1-form $A$, so the Yang-Mills equation is $delta dA=0$, but considering the inner product $langle-,-rangle$ we have for all 1-form $B$ $$0=langle B,delta Frangle=langle B,delta dArangle=langle dB,dArangle$$ in particular for $B=A$, so $$0=langle dA,dArangle Longrightarrow 0=dA=F,$$ i.e., if a connection satisfies the Yang-Mills equation then its curvature is zero. But if this is correct, in $mathbb{R}^4times U(1)$ we have that the electromagnetic field is always zero (F=0) which is clearly wrong. What is going on?? I can’t see it!! Is this apparent contradiction because in $mathbb{R}^4$, $langle-,-rangle$ is not defined for all 1-forms?? or because it is not an inner product (taking the Minkowski metric)??