I am a bit confused about a calculation I did and I do not see my mistake. First of I took any vector field $X \in \mathfrak{X}(M)$ and lifted it as some $\hat{X}\in \mathfrak{X}(T^*M)$ (s.t. $T\pi\hat{X}=X$). I calculated that it satisfies $$ \mathfrak{L}_\hat{X}\Theta=0 $$ for $\Theta \in \Omega^1(T^*M)$ the canonical 1-form and I checked this here Cotangent lift of the commutator of 2 vector fields
Here comes my confusion: If I have $X \in \mathfrak{X}(M)$ I can always get $fX \in \mathfrak{X}(M)$ for $f \in C^{\infty}(M)$ some function. And $\hat{fX}=f\hat{X}$ will be a possible lift of $fX$. Then I calculated using Cartan' magic formula: $$ \mathfrak{L}_{\hat{fX}}\Theta=i_{f\hat{X}}d\Theta+di_{f\hat{X}}\Theta =fi_{\hat{X}}d\Theta+d(fi_{\hat{X}}\Theta)=fi_{\hat{X}}d\Theta+df\cdot i_{\hat{X}}\Theta+f\cdot di_{\hat{X}}\Theta=f\mathfrak{L}_\hat{X}\Theta+df\cdot i_{\hat{X}}\Theta $$ But now using $\mathfrak{L}_{\hat{X}}\Theta=0$ we get $$ \mathfrak{L}_{\hat{fX}}\Theta=df\cdot i_{\hat{X}}\Theta $$ But this cannot be, since also $\mathfrak{L}_{f\hat{X}}\Theta=0$ should hold. And one can certainly finds $f\in C^{\infty}(M)$ such that $df \neq 0$. And $i_\hat{X}\Theta \neq 0$ as well for general lifted $\hat{X}$ when one uses in local coordinates $\Theta=p_idq^i$. It has to be something stupid and maybe someone can help me.