Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, $\mathbb{C}[\mathfrak{g}]^G$ is a polynomial algebra with generators $P_1,\cdots, P_n$ of degrees $d_1,\cdots, d_n$.
Even though the generators $P_i$ are not unique, sometimes there are nice choices. For instance, in type $A$, we can take the coefficient of the characteristic polynomial. Let us denote these invariant polynomials by $c_1,\cdots, c_n$.
Question: Are there analogues of these polynomials for general types? Said differently, can we give a purely root theoretic definition of the $c_i$'s, one which produces an invariant polynomial (of degree $d_i$) for every root system (of rank $n$)?
Note: Of course some choices/variants of the coefficient of characteristic polynomials also give us invariant polynomials in other classical types. But I'm looking for a construction that works uniformly for all types.
