Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space $V$. The action $\sigma : G \times V \to V$ induces the differential $d\sigma: Lie(G) \times V \to V$.
If $Lie(H)$ preserves a subspace $W \subset V$ (that is, $d\alpha(Lie(H) \times W) \subset W$), must $H$ also preserve $W$ (that is, $\sigma(H \times W)\subset W$)?
And if there are counter-examples, what further assumptions on $H$ and $G$ would make this statement true?