All groups are linear algebraic over some fixed field $k$.

I believe that it is true that, in characteristic $0$, if $G'$ is a reductive subgroup of $G$, then there is a $G'$-invariant complement to $\operatorname{Lie}(G')$ in $\operatorname{Lie}(G)$. I guess this is nearly a consequence of complete reductivity, but I'm not sure how to ensure rationality of the complement if some isotypic component for the adjoint component of $G'$ on $\operatorname{Lie}(G)$ is neither contained in, nor disjoint from, $\operatorname{Lie}(G')$.

In positive characteristic, this can fail. I think, but haven't checked, that the adjoint embedding $\operatorname{PGL}_2 \to \operatorname{GL}(\mathfrak{pgl}_2)$ in characteristic 2 is a counterexample.

Now, there are some phenomena that behave well in zero or large characteristic, but that fail in small characteristic, like the existence of Jacobson–Morosov triples. (EDIT: @JimHumphreys points out that this example may be misleading.) In this setting, we have McNinch's theory of optimal $\operatorname{SL}_2$-homomorphisms, of which it is my understanding that they often provide a substitute for JM triples that allow the use of many of the classical techniques.

My question is whether there is some similar substitute for the existence of complements to reductive Lie subalgebras in small positive characteristic. A specific example that would work for me is if it were known that, whenever $G'$ is a reductive subgroup of $G$, then $\operatorname{Lie}(G')$ surjects onto $\mathrm N_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))/\mathrm C_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))$. (This is a consequence of, but hopefully strictly weaker than, the existence of a complement.) In case it makes the problem any easier, it would suffice for me to be able to handle the case where $G'$ is adjoint, $G$ is $\operatorname{GL}(\operatorname{Lie}(G'))$, and the map $G' \to G$ is the adjoint embedding.

EDIT: @JimHumphreys points out that it may be unclear exactly what I am asking. First, my reference to "small positive characteristic" is misleading; I am looking for something that works in *all* characteristics. At least part of the question is "what is the 'right' replacement?" (to which one can imagine that there are many possible answers). One awkward, but sufficient for my purposes, replacement is:

If $G'$ is an adjoint group, $G = \operatorname{GL}(\operatorname{Lie}(G'))$, and $G' \to G$ is the adjoint embedding, then $\operatorname{Lie}(G')$ surjects onto $\mathrm N_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))/\mathrm C_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))$.

and so a proof of this (or counterexample) would certainly constitute an answer.