# Dual Space to a Lie Algebra. Problem from Takhtajan. Finding the center of a Poisson Algebra

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.