So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups).
Initially I saw their description as group objects in the category of affine schemes over $k$, which is also equivalent to a group object in the category of functors $\mathit{Alg}_k \to Set$ which is representable, which is also equivalent to a Hopf algebra.
When getting deeper into the theory, namely learning about parabolic and Borel subgroups, many texts started working with what I believe is $|G|$, the closed points of $G$ over the algebraic closure $\overline{k}$, sometimes denoted $\mathrm{spm}(G_{\overline{k}})$. This has the structure of a semi-topological group which seems useful in proving facts about $|G|$. My question is, are there general ways to lift results about $|G|$ to results about $G$?
For example, there is a neat result that for a group homomorphism $\phi: G \to H$ the corresponding map $|G| \to |H|$ has closed image, and I vaguely remember that $\mathrm{Im}(\phi)$ itself is closed in $H$ (correct me if I am wrong), and I wonder if you can prove the latter from the first.
What I tried
I became more interested in the question "how much is the functor of points of a scheme determined by its values on (algebraically closed) fields" but I could not find anything online even after a thorough search, so I produced the following result, which may still be wrong.
Proposition: Let $F,G: X \to Y$ be distinct morphisms of schemes with $X$ reduced with finitely many irreducible components. Then there is a field $L$ so that the functor of points morphisms $F(L),G(L): X(L) \to Y(L)$ are distinct.
Here we can obviously replace $L$ with an algebraically closed field. This would be helpful in lifting any result which can be stated in terms of a comparison between two morphisms.
Disclaimer: This may have been better to post on stackexchange, and I haven't posted any questions in years on either site. However I wanted the opinion of someone who has worked with algebraic groups for a while.