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).



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> 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 $E\_8$?
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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 $E\_8$.



Polynomial invariants of the exceptional Weyl groups
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 $E\_8$](http://jmp.aip.org/resource/1/jmapaq/v22/i9/p1857_s1). For the case of $G\_2$, see the paper in my comment above: [Casimir operators for the exceptional group $G\_2$](http://arxiv.org/abs/hep-th/9306062) by A.M. Bincer and K. Riesselmann.


