So I am trying to solve Problem 2.19 from the book "Quantum mechanics for mathematicians" by Takhtajan. The problem is the following:

Let $g$ be a finite-dimensional Lie Algebra with a Lie bracket $[,]$, and let $g^{*}$ be its dual space. For $f,h \in C^\infty(g^{*})$ define \begin{align*} \{f,h\}(u) = u([df,dh]) \end{align*} where $u \in g^{*}$ and $T^{*}_ug^{*} = g$. ... Determine the center of $C^\infty(g^{*})$. The center of $C^\infty(g^{*})$ is \begin{align*} \{f \in C^\infty(g^{*}): \{f,h\} = 0 \quad\forall h \in C^\infty(g^{*})\} \end{align*} It is also stated that the center does not consist only of locally constant functions.

The following should be true: If $\{f,h\}=0$ then $[df,dh]=0$. But now I don't know how to solve this equation, i.e. finding a function that is not locally constant with this property. Thank you for your advice.


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