# Lifting \$mathfrak{sl}_2\$-triples

Let

• $$k$$ be an algebraically closed field,
• $$G$$ a (smooth, connected) reductive algebraic group over $$k$$,
• $$H$$ a (smooth, connected) reductive group of semisimple rank 1, and
• $$T$$ a maximal torus in $$H$$.

I am specifically interested in the case where the characteristic of $$k$$ is a bad prime for $$G$$.

Suppose that we are given a group embedding $$F_T : T to G$$ and a $$T$$-equivariant Lie-algebra embedding $$f_H : mathfrak h to mathfrak g$$, such that the restrictions to $$mathfrak t$$ of $$f_H$$ and the derivative of $$F_T$$ agree. Can we extend $$F_T$$ to an embedding $$H to G$$ whose derivative is $$f_H$$?

(I know that most results in this area are stated under the assumption of good characteristic, or even of characteristic 0, but I don’t actually know any of the counterexamples or where to look for them, so I can’t tell if the hypothesis of the existence of $$T$$ is strong enough to allow me to overcome bad characteristic.)