TITLE: Failure of Jacobson-Morozov in positive characteristics
QUESTION [8 upvotes]: The Jacobson-Morozov theorem that any nilpotent element $e$ in the Lie algebra of a simple algebraic group $G$ can be embedded in an $\mathfrak{sl}_2$-triple, has a restriction (in terms of the Coxeter number) on the characteristic of the underlying field (assumed to be algebraically closed). This restriction is also required for the "uniqueness" of the triple, up to $C_G(e)$-action. (This result is due to Kostant.) In his 1980 paper, Pommerening had removed the restriction on the characteristic in Jacobson-Morozov's theorem, up to very small exceptions (i.e., characteristic is "bad"). Does the uniqueness as in Kostant's result also hold with this weaker restriction? If it does, then where does Jacobson-Morozov along with uniqueness result of Kostant fail in positive characteristics?

REPLY [7 votes]: Sorry to necro-post, but I have been meaning to point out for a while that I have a reasonably complete answer to the OP in some work with Adam Thomas, 'The Jacobson--Morozov Theorem and complete reducibility of Lie subalgebras' just made available in Proc. LMS, here. We were interested in $G$-complete reducibility of subalgebras of the Lie algebra $\mathfrak g$ of the reductive algebraic group $G$. A subalgebra is $G$-completely reducible if whenever it is in a parabolic subalgebra $Lie(P)$ of $\mathfrak g$ it is in a Levi subalgebra $Lie(L)$ of $Lie(P)$, a definition essentially due to Serre. It turned out that this definition matches up nicely with the Jacobson--Morozov theorem in that $G$-complete reducibility of all subalgebras guarantees you the uniqueness. We had to do a lot of case-by-case analysis, unfortunately, but we did get out the following:
Let $h(G)$ be the Coxeter number of $G$, $p$ the characteristic of the algebraically closed field $k$.

Any nilpotent element $e$ can be extended to an $\mathfrak{sl}_2$ triple $(e,h,f)\in \mathfrak{g}\times \mathfrak{g}\times \mathfrak{g}$ uniquely up to simultaneous conjugation by $C_G(e)$, if and only if $p>h(G)$.

Let $b(G)$ be the largest prime amongst those for which the Dynkin diagram contains an $A_{p-1}$ subdiagram and bad primes for $G$.

The number of nilpotent orbits and conjugacy classes of $\mathfrak{sl}_2$-subalgebras is the same if and only if $p>b(G)$. Moreover a bijection can be realised in a natural way by sending an $\mathfrak{sl}_2$-subalgebra to the nilpotent orbit of largest dimension meeting it.

The quantitative difference between the two results above essentially arises from the qualitative fact that when $p\leq h(G)$, then an $\mathfrak{sl}_2$-subalgebra can contain two non-conjugate nilpotent elements.
We also established, using the classification of nilpotent orbits, for the bad primes $p=3$ and $p=5$ in the exceptional groups, which nilpotent elements can be extended to $\mathfrak{sl}_2$-triples. All can except the exceptional orbit with label $A_2^{(3)}$ in $G_2$ when $p=3$.