I'm reading through Milne's book on algebraic groups, and in corollary 6.19 he writes:
Assume that $G$ is affine or smooth, then (c) for all $k-$algebras $R$, $(\mathcal{D}G)(R)$ consists of the elements of $G(R)$ that lie in the derived group of $G(R')$ for some faithfully flat $R$-algebra $R'$
He says this is an immediate consequence of the following proposition:
If $G$ is affine or smooth, then $\mathcal{D}G$ is the algebraic subgroup of $G$ generated by the commutator map $G\times G \rightarrow G$, $(g_1,g_2) \rightarrow [g_1,g_2] = g_1 g_2 g_1^{-1} g_2^{-1}$
The issue is, I don't really see why the corollary would follow so easily from this proposition. Any help would be appreciated.
P.S: I am aware there is an argument using generic flatness for the above claim, but it only applies for smooth schemes and besides I'm interested specifically in understanding what Milne meant in his argument.