Let $\mathfrak{g}$ be a simple complex Lie algebra. Let us fix a Cartan, a Borel, and generators $x_\alpha$ of negative simple roots. Then $N:=\sum x_\alpha$ is a principal (=regular) nilpotent element. Now suppose $X$ is an element of the Cartan.
Question: What is the Jordan decomposition of $N+X$?
In Type $A$, one finds that $N+X$ is conjugate to $N'+X$ where $N'$ is the principal nilpotent of the centraliser $C_{\mathfrak{g}}(X)$. The expression $N'+X$ is then the Jordan decomposition of $N+X$. (Since $N'$ is nilpotent, $X$ is semisimple, and they commute). Does something similar happen in general types?
