You can also see the book by Jantzen to find what Professor Premet said.
The book is Nilpotent orbits in representation theory pages 1-221, in ''Lie
Theory, Lie algebras and representations'', Progress in Mathematics
228, Birhauser, Boston.Basel.Berlin, 2004.

You can find it in Proposition 2,page26 of above book.
Maybe that will give you a precise explanation about the structure of Centralizer of nilpotent element.