Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the complexification. Denote by $K_\mathbb{C}$ the analytical Lie group corresponding to $\mathfrak{k}$. Then $K_\mathbb{C}$ acts on $\mathfrak{p}$. If $\mathfrak{g}_0$ is not of Hermitian type, then $K_\mathbb{C}$ acts irreducibly on $\mathfrak{p}$. Fix a Cartan subalgebra and a positive system for $K_\mathbb{C}$, and let $X$ be a highest weight vector in $\mathfrak{p}$ as the $K_\mathbb{C}$-representation. It is well known that the minimal orbit of the $K_\mathbb{C}$-action on $\mathfrak{p}$ is $K_\mathbb{C}\cdot X$.
Is there any other canonical representative of this minimal orbit? Namely, is there any other element $Y\in\mathfrak{p}$ (which is easy to write down) such that $K_\mathbb{C}\cdot Y=K_\mathbb{C}\cdot X$?
