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I am looking for a proof or a reference of the following claim:

Let $G$ be a real connected semi-simple Lie group and let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition for its Lie algebra. Let $M$ be a closed adjoint orbit in $\mathfrak{g} $. Then there exists a unique $(\beta_0, \beta_1) \in \mathfrak{k} \times \mathfrak{p}$ with $[\beta_0,\beta_1]=0$ such that $M= G. (\beta_0+\beta_1).$

  1. Does a similar property also hold for an orbit $M$ in the Lie algebra of a compact Lie group which is endowed with an involution $\theta$, namely does there exists a unique $(\beta_0, \beta_1) \in \mathfrak{g}^\theta \times \mathfrak{g}^{-\theta}$ such that $M= G.(\beta_0+ \beta_1)$ ? ( $\mathfrak{g}^{\theta}$ and $\mathfrak{g}^{-\theta}$ are the subspaces fixed by the induced involution on $\mathfrak{g}$ and its opposite respectively ) .
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    $\begingroup$ What is the property you want for a compact Lie group $G$? In that context, $\mathfrak k = \mathfrak g$ and $\mathfrak p = \{0\}$, so the decomposition is easy. (Also, in that context, every orbit is the continuous image of $G$, hence compact, hence closed in $\mathfrak g$.) $\endgroup$
    – LSpice
    Commented Mar 16, 2022 at 1:24
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    $\begingroup$ Yes. Still 'closed' is redundant for an orbit of a compact group, but of course there is no harm in stating it anyway. $\endgroup$
    – LSpice
    Commented Mar 16, 2022 at 13:30
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    $\begingroup$ I believe the following implies what you need: for any finite order automorphism $\theta$ of a semisimple Lie algebra one can find a Cartan subalgebra that meets generically each of the eigenspaces of $\theta$ (this follows from Vinberg's theory of theta-groups) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 16, 2022 at 14:49
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    $\begingroup$ I agree with @მამუკაჯიბლაძე that this must be a question about $\theta$-groups (I was just coming here to mention Kostant and Rallis - Orbits and representations associated to symmetric spaces), but the papers that I know seem mainly to talk about $K$-orbits on $\mathfrak p$. @‍მამუკაჯიბლაძე, do you have a reference for your result? (Probably it's in the Kostant–Rallis paper, or in one of Vinberg's; I don't know them well.) $\endgroup$
    – LSpice
    Commented Mar 16, 2022 at 15:01
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    $\begingroup$ @LSpice In this generality - Vinberg, "Weyl group of a graded Lie algebra" (paywall, sorry). He has detailed analysis of closed orbits, notably the notion of Cartan subspace in an eigenspace of a finite order automorphism, and many other goodies. I don't see all the details though, that's why I am commenting rather than answering. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 16, 2022 at 15:28

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