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The many comments as well as the answer by zroslav probably add to the confusion resulting from the original unfocused formulation of the question. First, it's not really a question about Lie groups …

One way to approach both the general and the specialized questions you raise is to study these Lie groups as $\mathbb{R}$-points of complex semisimple algebraic groups defined over $\mathbb{R}$. In …

This is a long comment rather than an answer, since I don't understand precisely what the question really means. I'm still confused about the formulation here, though Knop and others have contri …

The term "half-spin" seems to be more widespread in the literature. Anyway, the groups have serious uses in the study of $E_8$ and other exceptional Lie algebras (by people including Skip Garibald …

To amplify Ben's answer, I'd point to an earlier post that has lots more detail: here. The subject of compact groups is old and well-studied, so there are many references to choose from, even Wiki …

The two modules in the question are actually isomorphic, since highest weights determine irreducibles. Anyway, the answer to your question is that there is usually no explicit decomposition known. …

Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalle …

In the direction of a positive answer to your question, there are some rather restrictive hypotheses which would suffice, though I'm still not optimistic about getting a positive answer in your very g …

I'm still unsure what you are looking for (or where you saw the material you recall), but the work of Borel-Tits and also Satake on reductive groups over arbitrary fields including $\mathbb{R}$ might …

Rather than imitate Matt Emerton's multiple comment style, I'll just make my answer community-wiki (since there is much to say from different viewpoints). The first lesson is that you have to insist …

It may be useful to expand my comments. The question involves Lie type $C_m$ with $m \geq 2$. Without developing Lie group or algebraic group language, it's enough to work with a simple Lie algebr …

The answers by Will and Paul, along with various comments, are probably enough to convince anyone that the question asked is exceptionally broad (too broad for a self-contained answer) and involves a …

Probably it's more natural to talk about the Jordan decomposition and regularity when the groups are interpreted as semisimple algebraic groups (or real forms thereof). I don't think there is a spec …

To supplement algori's answer: The first interesting case (for the Lie algebra of type $A_1$ over $\mathbb{C}$) is already noted in the question: the odd dimensional irreducible representations expone …

To follow up Skip's comment, assuming the underlying field is algebraically closed of characteristic 0, the number of possible embeddings (= injective homomorphisms) of $\mathfrak{s}\mathfrak{l}_2$ in …