Let $G$ a matrix Lie group and $\mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $\mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true and can be seen via the adjoint representation and the exponentiation map. Does the same holds when $G$ is an affine linear algebraic group over a field $K$, $\text{char}(K)>0$? If yes, how can I see this?
I do not have much experience with both Lie and algebraic groups. Please be gentle :)
EDIT: By conjugation I mean the action of the adjoint representation. In the matrix case this is equal to conjugation by an elemeng $g\in G$ and stable subspaces under this action seem to be ideals, at least for classic Lie groups (manifolds). But what if $G=G(\mathbb{F}_q)$ is a linear algebraic group?