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 ?