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First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include …

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 …

The problem with your highlighted formulation is that it's wrong as stated, unless for example you require that an "irreducible" representation be finite dimensional or have an integral highest weight …

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 …

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 …

To supplement what Pete says, I'd emphasize that "root systems" have been defined in a variety of ways for a variety of purposes related to Lie theory or to some type of "reflection" group. (For in …

The basic set-up here makes sense in the theory of abstract root systems if one brings (integral) weights into the picture, but it may be more natural to think about the classical characteristic 0 the …

I'd be extremely surprised if such tables or database existed, mainly because the number of possible reduced decompositions for a Weyl group element tenda to grow very large as the rank increases. …

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 …

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 the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis …

The significance of $\rho$ (and the associated dot-action of the Weyl group or other Coxeter group) in representation theory is discussed from many angles in the earlier linked question. When deal …

I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case. It's probably best to start with an irreducib …

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 …

As my comment suggested, the question itself lacks detail and seems to be out of focus. For the somewhat intricate structure theory of a (real) simple or semisimple Lie group, there are standard tex …