Given a nilpotent orbit, $\mathcal{O}_X$, associated to an element, $X$, of a complex semisimple Lie algebra, $\mathfrak{g}$, there are two equivalent classification schemes:
Via the Jacobson-Morozov theorem, one can define a triplet $(H, X, Y)$ satisfying the usual $\mathfrak{sl}_2$ commutation relations, such that $\alpha(H)\in\{0,1,2\}$ for $\alpha$ a simple root of $\mathfrak{g}$. This means that to any nilpotent orbit one can associate a unique weighted diagram $D(\mathcal{O}_X)$ where each of the nodes are labelled by 0,1,2.
Via Bala-Carter theory for each nilpotent orbit one associate a pair (up to conjugacy) $(\mathfrak{l},\mathfrak{p}_\mathfrak{l})$ where $\mathfrak{l}$ is a Levi subalgebra and $\mathfrak{p}_\mathfrak{l}$ a distinguished parabolic subalgebra of $[\mathfrak{l}, \mathfrak{l}]$.
Both approaches are standard, but I have not found a reference explaining how to go from one to the other. Is there a simple way to read off the Levi subalgebra from a weighted Dynkin diagram? Looking at the tables for exceptional algebras in [1] where both are given, I cannot see how to go from the diagrams to the BC labels. I couldn't find an answer for classical series either, where the orbits are usually labelled by partitions/Young tableaux.
[1] Collingwood and McGovern, Nilpotent Orbits in Semisimple Lie Algebras
