Let $G$ be a simple linear group group over an algebraically closed field $k$, and let $B$ be a maximal solvable subgroup.
If things are happening over $\mathbb{C}$ then I know how to show that $B$ is not nilpotent: elements of torus lie subalgebra of $B$ act in a non-nilpotent way via adjoint representation.
But is there an equivalence between Lie algebra being nilpotent and Lie group being nilpotent in general, when there is no exponential map? what does one do in positive characteristic?
So the question is: how to show that $B$ is not nilpotent when $k \neq \mathbb{C}$?