  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 $\mathfrak{g} := \text{Lie}(G)$ and also those
with overalgebras isomorphic to the algebras $\text{Lie}(\text{SL}_2)$ and
$\text{Lie}(\text{PGL}_2)$. This leads us to calculate the dimension of Lie
automiser $\mathfrak{n}_\mathfrak{g}(k\cdot e)/\mathfrak{c}_\mathfrak{g}(e)$
for all nilpotent orbits; in even characteristic this quantity is very
sensitive to isogeny.
