Let $G$ be an algebraic group over $k$ which is either smooth or affine. Then the derived group can be defined by the algebraic subgroup generated by the image of the morphism $$G\times G \rightarrow G, (g,h)\mapsto ghg^{-1}h^{-1}$$ i.e. the minimal algebraic subgroup containing the image of the above morphism.
In the note Algebraic groups, The theory of group schemes of finite type over a field. by Milne, 8.21(c), he says that $[G,G](R)$ consists of elements of $G(R)$ that lie in $[G(R'),G(R')]$ for some faithfully flat R-algebra $R'$.
The proof given in the note does not give the detail. Can anyone prove this proposition?