Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on the Wikipedia page for their explicit values).

If a positive integer $d$ is a degree of $W$, then so is $h+2-d$, where $h$ is the Coxeter number associated to $W$. This suggests considering the real polynomial $$P_W(x) := \prod_{j=1}^n \left(x - \exp\left(2\pi i\frac{d_j}{h+2}\right)\right).$$

For instance, in type $G_2$, the degrees are $2$ and $6$, $h+2 = 8$ and so $$P_{G_2}(x) = x^2+1.$$

Precise question: Is there a reference in the literature for the expanded form of these polynomials for all types $ABCDEFG$?

Vague question: Has anyone studied such polynomials? In particular, is there a way of seeing them arise naturally (e.g. as characteristic polynomials, or other…)?