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It is shown that Suzuki-Ree groups can be easily defined by means ofcomparing two fundamental representations of the ambient Chevalley group incharacteristic 2 or 3. This eliminates the distinction between the Suzuki-Reegroups over perfect and imperfect fields and gives a natural definition for theanalogues of such groups over commutative rings. As an application of the sameidea, we explicitly construct a pair of polynomial maps between the groups oftypes B_n and C_n in characteristic 2 that compose to the Frobeniusendomorphism. This, in turn, provides a simple definition for the Tits mixedgroups over rings.

Suzuki-Ree groups and Tits mixed groups over rings