Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions:
$e$ is principal in some Levi subalgebra $\mathfrak l$ of $\mathfrak g$.
There exists a good even grading on $\mathfrak g$ (recall that a grading on $\mathfrak g$ is called good e if $e \in \mathfrak g_2$ and the linear map $ad~ e : \mathfrak g_j → \mathfrak g_{j+2}$ is injective for $j \leq −1$ and surjective for $j\geq −1$).
My question is this: is there any relation between these conditions, or are they completely independent?