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May 2, 2018 at 13:35 comment added Robert Bryant @IgorBelegradek: YCor answered the question you actually asked, and that answer was first, so you made the correct choice.
May 2, 2018 at 13:07 comment added Igor Belegradek I love both answers but can only accept one.
May 2, 2018 at 11:45 comment added Igor Belegradek (cont.) except possibly if $n=3$ where one has to deal with $G_2$ structure group.
May 2, 2018 at 11:30 comment added Igor Belegradek Thank you, in the weaker question I am also asking about Lie algebras of compact Lie groups. In my application they appear as structure groups of $SO(2n+1)$ principal bundles. I wanted to know what can be said about the bundle if the structure group is a closed proper (!) subgroup of $SO(2m+1)$. From the answer to the weaker question it follows that for bundles with smaller structure group the top rational Pontryagin class is a polynomial in lower-dimensional Pontryagin classes and the Euler class, which (potentially) restricts the bundle.
May 2, 2018 at 11:17 comment added YCor Thanks (sorry I didn't read "of compact type")
May 2, 2018 at 11:12 comment added Robert Bryant @YCor: Well, I would imagine that "simple Lie algebra over $\mathbb{R}$ of compact type" does imply "real form". Since the original question was about closed subgroups of $\mathrm{SO}(2n{+}1)$ which are compact, I feel that it is justified to restrict to simple Lie algebra over $\mathbb{R}$ of compact type, and the 'weaker' question seems to be about simple subalgebras of ${\mathfrak{so}}_{2n+1}$, which are also of this type. Hence my rephrasing of the question in the form that I did.
May 2, 2018 at 11:05 comment added YCor @RobertBryant you mean, all real forms of these types?
May 2, 2018 at 11:02 comment added Robert Bryant @YCor: When one phrases the question as "Which simple Lie algebras over $\mathbb{R}$ of compact type have the property that their nontrivial representation(s) of lowest dimension have odd dimension?", then, yes, the answer is only ${\mathfrak{so}}_{2n+1}$ and ${\mathfrak{g}}_{2}$.
May 2, 2018 at 8:31 comment added YCor It might me the only exception? I guess the answer is yes for large $n$.
May 2, 2018 at 0:50 history edited Robert Bryant CC BY-SA 4.0
Fixed a typo, added a reference
May 1, 2018 at 23:05 comment added paul garrett Haha! A crazy fact! :)
May 1, 2018 at 22:36 history answered Robert Bryant CC BY-SA 3.0