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.