Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the intersection of $\mathcal{O}$ with Levi subgroups (or probably, their derived groups) and Pseudo-Levi subgroups. I guess that there is a way to take the weighted Dynkin diagram of $\mathcal{O}$ (or the nilpotent orbit associated to it in the Lie algebra) and construct the weighted Dynkin diagram of the intersection.

As an example to what I'm trying to figure out, I went to Birne Binegar's UMRK database (at http://umrk.dynns.com:800/UMRK/UMRK.html) and looked at the data for Nilpotent orbits of the Lie algebra of type $E_7$. For the orbit $A_4+A_1$ it says that the producing pseudo-Levi subalgebra is generated by the roots [0, 1, 2, 3, 5, 6, 7] (so the negative highest roots together with roots number 1,2,3,5,6,7 in the Dynkin diagram, enumeration as in Bourbaki I believe). The producing pseudo-Levi subalgebra means here (as far as I understand) a standard pseudo-Levi subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ such that the intersection of $A_4+A_1$ with $\mathfrak{h}$ is distinguished. My question is, what orbit of $\mathfrak{h}$ is the intesection? In particular, I wish to know if and how can I find a subalgebra such that the intersection is the principal orbit.

A closely related follow-up question is, can I then go about determining the stabilizer of the orbit from this data (say, the stabilizer of the intersection in $\mathfrak{h}$ and somethineg like $\mathrm{Aut}(\mathrm{Dyn}(\mathfrak{h}))$?

A bonus question, if someone happens to know: The UMRK attaches a further integer to the pseudo-Levi subalgebra on top of the list of roots (it's 4 in the case), what does this integer represents?