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ADDED: I'd emphasize that writing suitable generators (Casimir operators) for the center of $U(\mathfrak{g})$ should involve a choice of PBW or other basis, though the initial approach might not start with such a basis but rather with the Killing form. However this is done (non-uniquely), it takes some care to realize from these operators a set of basic polynomial invariants for the Weyl group. The latter calculation by itself can be done much more straightforwardly, though hardly anyone has taken the trouble to write down (for example) a basic invariant polynomial in 8 variables of degree 30 for $E_8$. The 1988 paper by M.L. Mehta in Communications in Algebra seems to be a good attempt at giving a comprehensive treatment. Unfortunately, the journal itself is not so easy to access, and my own copy of the paper photographed from typescript by the journal is barely readable.

I have had less success in deciphering the physics literature, which may or may not all be mathematically reliable. In particular, I haven't yet reached any conclusions about what is in the JMP paper cited by Jose. (That journal is sometimes quite useful but can also be quite frustrating to extract information from for mathematical purposes.) My only experience has been with the literature on finite (mostly real) reflection groups and their invariants, where the degrees themselves are most important for most applications. One concrete source I should mention is the added Chapter 7 in the second edition of Grove-Benson Finite Reflection Groups (GTM 99, Springer, 1985). Their book was first developed as an advanced undergraduate text, then expanded somewhat, and gives more details than my book --- where for instance I left the computation of basic invariants for dihedral groups as an exercise.

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Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature. Those I was aware of before 1990 are listed in the references to section 3.12 of my book, but there may have been others I overlooked. A later paper of interest discusses "canonical" choices of generators, with details for the classical cases as well as dihedral groups (including $G_2$):

MR1469638 (98j:13007) 13A50 (20F55), Iwasaki, Katsunori (J-KYUS), Basic invariants of finite reflection groups. J. Algebra 195 (1997), no. 2, 538–547.

Physicists usually look for very explicit expressions, though their notation and approach may be hard for mathematicians to decipher. Their interest comes from the direction of Casimir operators as Jose points out in his literature citations. But those operators live in the center of the universal enveloping algebra, which by Harish-Chandra is isomorphic to the Weyl group invariants asked about here. The complication is that expressions for Casimir operators get much more elaborate-looking in terms of the Lie algebra notation. (Also, the reflection group theory shows that polynomial invariants and degrees play a uniform role even in non-crystallographic cases like $H_3$ and $H_4$ as well as dihedral groups which are not Weyl groups.)