I have a question regarding Cartan involutions of su(n). Some sources say that there is only one up to equivalence (Wikipedia on Cartan Decomposition). Others say there are Types I, II, III. I looked at Helgason's book, and he has the involution types I, II, III listed in Chapter 10. Could someone please clarify if these are all Cartan?
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1$\begingroup$ Any two Cartan involutions are equivalent in the sense, that they differ only by an inner automorphism. On the other hand, there are three types, AI, AII, AIII for $\mathfrak{su}(n)$ up to conjugation. $\endgroup$– Dietrich BurdeJun 12, 2013 at 17:53
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2$\begingroup$ If $\mathbb{su}(n)=\mathbb{p}+\mathbb{k}$, then the types are AI, AII, and AIII respectively, corresponding to the cases $\mathbb{k}\simeq \mathbb{so}(n)$, or $\mathbb{k}\simeq \mathbb{sp}(n/2)$, or $\mathbb{k}\simeq s(\mathbb{u}(p)+\mathbb{u}(q)$ with $p+q=n$. $\endgroup$– Dietrich BurdeJun 12, 2013 at 18:22
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$\begingroup$ How can all Cartan involutions be equivalent by inner automorphisms? Don't inner automorphisms conserve the spectrum as well? Could you give an explicit example of one which doesn't? e.g. on su(n) how are $\theta(X)=X$ and $\theta(X)=X^*= -X^T$ related? $\endgroup$– Another UserJan 30 at 14:02
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