Let $\mathfrak{g}$ be a real semisimple Lie algebra. A subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is *parabolic* if its complexification is parabolic in $\mathfrak{g}_\mathbb{C}$, meaning it contains a Borel subalgebra of $\mathfrak{g}_\mathbb{C}$. A nilpotent element of $\mathfrak{g}$ is *regular* or *principal* if the dimension of its stabilizer is minimal with respect to the stabilizers of all nilpotent elements of $\mathfrak{g}$. In "Wolf, Remark on nilpotent orbits", before Corollary 2 it is claimed that:

If $e \in \mathfrak{g}$ is regular-nilpotent, then $e$ is contained in a

uniqueminimal parabolic subalgebra of $\mathfrak{g}$.

**Question:** Is there an easy proof for this statement, or a reference?

(**Bonus:** Does this property uniquely characterize regular nilpotent elements, i.e. if a nilpotent element is contained in a unique minimal parabolic subalgebra, is it regular?)

No reference for this is provided, and I could also not find this claim in any of the sources of the article. In the complex setting, this is a standard statement which one can, for example, look up in Bourbaki -- remembering that "Borel = minimal parabolic". But I don't think the proofs are easy to adapt to the real setting, since the structure theory of real minimal parabolics seems a tiny bit more complicated than in the complex setting.

Lastly: If someone could recommend to me any further literature on the study of minimal parabolics in real semisimple Lie algebras, that would also be appreciated. At the moment, I'm mostly using Bourbaki's Chapter 7-9 and Knapp's "Lie Groups beyond an introduction".

*Bourbaki, Nicolas*, Elements of mathematics. Lie groups and Lie algebras. Chapters 7–9. Transl. from the French by Andrew Pressley, Berlin: Springer (ISBN 3-540-43405-4/hbk). xi, 434 p. (2005). ZBL1139.17002.

*Knapp, Anthony W.*, Lie groups beyond an introduction, Progress in Mathematics (Boston, Mass.). 140. Boston: Birkhäuser. xv, 604 p. (1996). ZBL0862.22006.beyond an introduction".

*Wolf, Joseph A.*, **Remark on nilpotent orbits**, Proc. Am. Math. Soc. 51, 213-216 (1975). ZBL0316.22013.