Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map $\Bbb C[{\frak g}]\to\Bbb C[{\frak h}]$ induces an isomorphism of graded $\Bbb C$-algebras $$\Bbb C[{\frak g}]^G\to\Bbb C[{\frak h}]^W.$$ What is the inverse of this map?
Using root-space decomposition, there is a projection $$\pi:{\frak g}={\frak h}\oplus\bigoplus_{\alpha\in\Phi}{\frak g}_\alpha\to {\frak h}.$$ Thus, we have a map $$\pi^*:\Bbb C[{\frak h}]\to\Bbb C[{\frak g}].$$ Is the restriction of this to $\Bbb C[{\frak h}]^W$ the inverse of the Chevalley restriction map?
No. Look at the matrix $\left(\begin{smallmatrix}0 & -1 \\ 1 & 0\end{smallmatrix}\right) \in \mathfrak{g}=\mathfrak{sl}(2)$. This is a sum of two elements of two root spaces using the standard Cartan. But the trace of the square is not zero, and trace of the square is an invariant polynomial.