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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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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. – Dietrich Burde Jun 12 '13 at 17:53
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 $p+q=n$. – Dietrich Burde Jun 12 '13 at 18:22

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