On extensions of the Jacobson-Morozov theorem to even characteristic

Let $G$ be a simple algebraic group over an algebraically closed field $\k$ of characteristic $2$. We consider analogues of the Jacobson--Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple $3$-dimensional Lie overalgebra in $\g:=\Lie(G)$ and also those with overalgebras isomorphic to the algebras $\Lie(\SL_2)$ and $\Lie(\PGL_2)$. This leads us to calculate the dimension of the Lie automiser $\n_\g(\k\cdot e)/\c_\g(e)$ for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.