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Jul 17, 2019 at 9:52 comment added AThomas Thank you YCor for the counter example. But still, I have the impression that for every Lie algebra $\mathfrak{g}$ there is a irreducible representation r such that equality holds in the right inclusion. For $\mathfrak{sl}_3$ it is the standard representation. I will edit the question.
Jul 16, 2019 at 18:34 history answered YCor CC BY-SA 4.0