Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}^*)$ be the ring of polynomial functions on $\mathfrak{h}$. The Weyl group $W$ acts on $\mathfrak{h}$, and this action extends to an action of $W$ on $S(\mathfrak{h}^*)$. It is a well-known fact that the space of Weyl group invariants $S(\mathfrak{h}^*)^W$ is generated by $r$ algebraically independent homogeneous generators, where $r$ is the dimension of $\mathfrak{h}$ (equivalently, the rank of $\mathfrak{g}$). The degrees of the generators are uniquely determined, though the actual generators themselves are not.
The degrees of the generators for $S(\mathfrak{h}^*)^W$ are well-known and can be found, for example, in Humphreys' book "Reflection groups and Coxeter groups" (Section 3.7). When $\mathfrak{g}$ is of classical type (ABCD), it is also not hard to find explicit examples of generators for $S(\mathfrak{h}^*)^W$ (loc. cit. Section 3.12).
Where, if anywhere, can I find explicit examples of generators for $S(\mathfrak{h}^*)^W$ when $\mathfrak{g}$ is of exceptional type, specifically, for types $E_7$ and/or $E_8$?
I have found explicit examples for types $E_6$ and $F_4$ in a paper by Masaru Takeuchi (On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 1962 313–328 (1962)). I have probably also come across examples for type $G_2$, though I don't recall where at this moment. But I have been unable to find anything for types $E_7$ or $E_8$.
Best Answer
In Physics these are called the "Casimir operators" and googling this gives the following paper: F. Berdjis and E. Beslmüller Casimir operators for $F_4$, $E_6$, $E_7$ and $E_8$. For the case of $G_2$, see the paper in my comment above: Casimir operators for the exceptional group $G_2$ by A.M. Bincer and K. Riesselmann.