Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are there any books that use the functor of points approach? Demazure-Gabriel's second book would have covered general group schemes in this way, but it was never written, and it's not clear whether or not it would have covered reductive groups anyway. There is a lot of material in SGA 3 using more modern machinery to study group schemes, ~~but I'm not aware of any significant treatment of reductive groups in that book ~~(although I haven't read very much of it).

Correction: Prof. Conrad has noted that SGA 3 does contain a significant treatment of reductive groups using modern machinery.

geometricnotion: over an imperfect field $k$ we cannot detect it using just $k$-subgroups. That is the whole point of the pseudo-reductive stuff. Functor of points is a convenient notion, even powerful for working with group objects and related constructions, but not everything can be shoehorned into it. Can you define (in a useful non-tautologous way...) the dimension of a finite type scheme over a field or quasi-compactness or irreducibility using functor of points? $\endgroup$ – BCnrd Mar 11 '10 at 4:46arefunctors by defn (no loc. ringed space), it is a tautology that any definition for alg spaces is literally expressed in terms of its "functor of points" since that's all it is! The crux is whether the defn/criterion is meaningful for a functor not known to be an algebraic space. For q-c (and irred) the answer is "no". I put "useful non-tautologous way" in my previous comment to head off a reply as you wrote. $\endgroup$ – BCnrd Mar 13 '10 at 18:02