Just curious, what does it mean for the adjoint action to be trivial on a commutative Hopf algebra $H$? Does it mean $\operatorname{Ad}_g(f)=\epsilon(g)f$, where $\epsilon$ is the counit, or $\operatorname{Ad}_g(f)=f$?
If $H$ is commutative as an algebra with antipode $S$, then I computed $$ \operatorname{Ad}_g(f)=\sum g_1fSg_2=\sum g_1Sg_2 f=\epsilon(g)f $$ so I suspect it's the former, but that doesn't seem like a truly "trivial" action like the latter.