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To comment on the question here (in community-wiki mode), I should point out first that $\S13$ of my now-ancient book was meant to develop some properties of weights just in the framework of abstract …

To compensate for my unfocused earlier comments it may be useful to supplement Richard's efficient answer based on Bourbaki's treatment of Coxeter elements in finite reflection groups. There is a sh …

This question (which I overlooked for a long time) reflects a natural notational confusion but is easy to answer. The Cartan integers themselves are unambiguous for each root system, but the meaning …

Maybe it's helpful to add a longer comment, in community-wiki format. The original question is not well-formulated, I think, as shown in the later convoluted remarks on the case $\theta =1$. It's p …

As indicated in my comments, the 1968 book Simple Groups of Lie Type by R.W. Carter has a good elementary treatment of your question (to which the answer is yes) in Chapter 13. All of this goes back …

After taking a closer look at the proof by Kac of Prop. 5.8 c), I can see that it's too sketchy to be followed easily. Here the generalized Cartan matrix is assumed to be of indefinite type, which I …

As others have pointed out, the word "lattice" in this context needs to be used with care. Leaving aside the use of this word to describe certain partially ordered sets, lattices in Euclidean space …

The role of the longest element in $W$ emerges only gradually in the Chevalley structure theory. This is developed similarly but in slightly different styles in the three books with the same title Li …

To me the question itself (and the answers) are out of focus, starting with the claim that the ring of Weyl group invariants is somehow central. Chevalley's 1955 argument does show that this ring is …

Bourbaki has the most detailed treatment, but they tend not to deal with weight lattices (or co-weight lattices) so explicitly outside their account of some of the representation theory. Thus you ca …

Apart from notation, the abstract root system argument is given at the end of section 9.4 in my 1972 textbook, Springer GTM 9. (See also section 8.4 for the origin in semisimple Lie algebras. Tog …

Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in Lie Groups and Lie Algebras. I'm not sure how strong t …

This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical ba …

The classification of simple Lie algebras (over $\mathbb{C}$ or other sufficiently large field of characteristic 0) correlates these Lie algebras with the irreducible reduced root systems (in Bourbaki …

Rather than prolong the tangled comments, I'll try to provide a straightforward answer to the current formulation of the question. As noted already, there are two small cases of irreducible root sy …