# Classification of real nilpotent orbits using real parabolic subalgebras?

Bala and Carter's approach to the classification of nilpotent orbits in complex semisimple Lie algebras $\mathfrak g$ is to associate to each nilpotent element $e\in\mathfrak g$ the pair $(\mathfrak l,\mathfrak q)$, where $\mathfrak l$ is a minimal Levi subalgebra of $\mathfrak g$ containing $e$ (unique up to $G_{ad}^e$-conjugation) and $\mathfrak q$ is the Jacobson-Morozov parabolic subalgebra associated to $e$. It turns out that $e$ is a so called distinguished element of $\mathfrak l'=[\mathfrak l,\mathfrak l]$ in the sense that the reductive part of the centralizer of $e$ in $\mathfrak l'$ is zero; equivalently, $\mathfrak q$ is a so called distinguished parabolic subalgebra of $\mathfrak l'$ in the sense that $\dim \mathfrak l_{\mathfrak q}=\dim \mathfrak u/[\mathfrak u,\mathfrak u]$ where $\mathfrak q=\mathfrak l_{\mathfrak q}\oplus\mathfrak u$ is the decomposition into reductive and unipotent parts, see the book by Collingwood and McGovern for details on this.

If $\mathfrak g_{\mathbb R}$ is a real form of $\mathfrak g$ with Cartan decomposition $\mathfrak g_{\mathbb R}=\mathfrak k_{\mathbb R}+\mathfrak p_{\mathbb R}$ and complexification $\mathfrak g=\mathfrak k+\mathfrak p$ , the Kostant-Sekiguchi correspondence maps nilpotent (adjoint) orbits of $\mathfrak g_{\mathbb R}$ to nilpotent $K$-orbits in $\mathfrak p$ via the so called Cayley transform. Using this correspondence, Noel adapted Bala and Carter's ideas to the real case by establishing a bijective correspondence between nilpotent $K$-orbits in $\mathfrak p$ and triples $(\mathfrak l,\mathfrak q, \mathfrak n)$ where $\mathfrak l$ is a minimal ($\theta$, $\sigma$)-stable Levi subalgebra of $\mathfrak g$ containing $e$ ($\theta$ is the Cartan involution and $\sigma$ is the conjugation of $\mathfrak g$ over $\mathfrak g_{\mathbb R}$), $\mathfrak q$ is a $\theta$-stable parabolic subalgebra of $[\mathfrak l,\mathfrak l]$ and $\mathfrak n$ is a certain $L\cap K$-prehomogeneous subspace of $\mathfrak q\cap \mathfrak p$ containing $e$. It turns out that $e$ is a so called noticed element of $\mathfrak l'$ in the sense that the reductive part of the centralizer of $e$ in $L\cap K$ is trivial.

Alternatively to Noel's approach, I have been wondering whether it makes sense and it is possible to carry out a classification of nilpotent orbits in real semisimple Lie algebras using minimal (real) Levi subalgebras and some class of real parabolic subalgebras?