I want to do exercise 10.1 of Waterhouse book which says that if an affine group scheme over a field isn't solvable then for some ring R it's R-point isn't solvable.
Waterhouse defines commutator subgroup this way: let $I_n$ be the kernel of the homomorphism $k[G]\to \otimes_{2n}k[G]$ associated to the map $G\times G\times ...\times G\to G$ sending $$(x_1,y_1,x_2,y_2,...,x_n,y_n)$$ to $$[x_1,y_1][x_2,y_2]...[x_n,y_n]$$ .define $DG$ as the group associates to $\cap I_n$ and he say G is solvable if for some $m$ we have $D^m G={e}$.
if G is finite-dimensional then I think we can reduce the problem to the case $DG=G$ and in this case, we can easily see that $$DG(k)=D(G(k))$$ but for general G I haven't any Idea.