Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ denote the Lie algebra of the maximal torus. Let $W=N_G(T)/T$ be the Weyl group. The Chevalley Restriction Theorem states that the the natural map $\mathbb{C}[\mathfrak{g}]\rightarrow\mathbb{C}[\mathfrak{t}]$ restricts to a $\mathbb{C}$-algebra isomorphism $$\mathbb{C}[\mathfrak{g}]^G\xrightarrow{\cong}\mathbb{C}[\mathfrak{t}]^W.$$ The latter is generated by $r=rk(G)$ algebraically independent polynomials $f_1,\ldots,f_r$.
Now, one has the root-space decomposition $$\mathfrak{g}=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}.$$ For each root $\alpha\in\Delta$, choose a non-zero root vector $e_{\alpha}\in\mathfrak{g}_{\alpha}$. If $\alpha\in\Pi$ is a (positive) simple root, set $h_{\alpha}:=[e_{\alpha},e_{-\alpha}]\in\mathfrak{t}$. Note that $$\{e_{\alpha}\}_{\alpha\in\Delta}\cup\{h_{\alpha}\}_{\alpha\in\Pi}$$ is then a basis of $\mathfrak{g}$. Let $$\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$$ be the associated dual basis of $\mathfrak{g}^*$.
$\textbf{Question}:$ Is there a way to choose the generators $f_1,\ldots,f_r$ so that each is nicely expressible as a polynomial in the indeterminates $\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$?
This question is rather vague, but I'm really just looking for explicit expressions of the $f_1,\ldots,f_r$ in terms of the $\{e_{\alpha}^*\}_{\alpha\in\Delta}\cup\{h_{\alpha}^*\}_{\alpha\in\Pi}$.
