Let $G$ be a simple algebraic group over the algebraically closed field $k$ of arbitrary characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is independent of good characteristic. There is quite a lot of history to the subject, but a uniform proof of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also independent of good characteristic? This fact seems to be used in *Varieties of nilpotent elements for simple Lie algebras I: good primes* by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.