Suzuki-Ree groups and Tits mixed groups over rings
It is shown that Suzuki-Ree groups can be easily defined by means of comparing two fundamental representations of the ambient Chevalley group in characteristic 2 or 3. This eliminates the distinction between the Suzuki-Ree groups over perfect and imperfect fields and gives a natural definition for the analogues of such groups over commutative rings. As an application of the same idea, we explicitly construct a pair of polynomial maps between the groups of types B_n and C_n in characteristic 2 that compose to the Frobenius endomorphism. This, in turn, provides a simple definition for the Tits mixed groups over rings.