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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 $M= G. (\beta_0+\beta_1).$

2. Does a similar property also hold for a closed 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 ) .

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 $M= G. (\beta_0+\beta_1).$

2. 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 ) .

I am looking for a proof or a reference for 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 $M= G. (\beta_0+\beta_1).$

Does this property also holds for a closed (or not) orbit in the Lie algebra of a compact Lie group ?

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 $M= G. (\beta_0+\beta_1).$

2. Does a similar property also hold for a closed 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 ) .

I am looking for a proof or a reference for 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 $M= G (\beta_0+\beta_1).$

Does this property also holds for a closed (or not) orbit in the Lie algebra of a compact Lie group ?

I am looking for a proof or a reference for 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 $M= G. (\beta_0+\beta_1).$

Does this property also holds for a closed (or not) orbit in the Lie algebra of a compact Lie group ?