The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G\_2(F)$ has an unusual automorphism of order $2$, which switches long root subgroups with short root subgroups. The fixed points of this automorphism, form a subgroup of $G\_2(F)$, which I think is called a Ree group.


A similar construction is possible, when $F$ is a perfect field of characteristic $2$, using Chevalley groups of type $B$, $C$, and $F$, leading to Suzuki groups. I apologize if my naming is not quite on-target. I'm not sure which groups are attributable to Suzuki, to Ree, to Tits, etc..


Unfortunately (for me), most treatments of these Suzuki-Ree groups use abstract group theory (generators and relations). Is there a treatment of these groups, as *algebraic groups* over a base field? Or am I being dumb and these are not obtainable as $F$-points of algebraic groups.


I'm trying to wrap my head around the following two ideas: first, that there might be algebraic groups obtained as fixed points of an algebraic automorphism that swaps long and short root spaces. Second, that the outer automorphism group of a simple simply-connected split group like $G\_2$ is trivial (automorphisms of Dynkin diagrams mean automorphisms that preserve root lengths). 


So I guess that these Suzuki-Ree groups are inner forms... so there must be some unusual Cayley algebra popping up in characteristic 3 to explain an unusual form of $G\_2$. Or maybe these groups don't arise from algebraic groups at all.


Can someone identify and settle my confusion?


Lastly, can someone identify precisely which fields of characteristic $3$ or $2$ are required for the constructions of Suzuki-Ree groups to work?



Suzuki and Ree groups, from the algebraic group standpoint
It is not really a question of inner forms. What happens is that the
*algebraic group* $G\_2$ has an extra endomorphism $\varphi$ whose square
is the Frobenius map (over the appropriate finite field). Just as for any
algebraic group over a finite field $F$ its rational points over $F$ are the
fixed points of the Frobenius endomorphism the Suziki groups are, by definition,
the fixed points of $\varphi$. Again, just as the Frobenius, on points over the
algebraic closure of $F$ $\varphi$ is an automorphism of the abstract group.
However, that is misleading, the essential points is that it is an endomorphism
(which definitely is not an automorphism) of the algebraic group. Most of the
properties of points over $F$ of a semi-simple algebraic group $G$ defined over
$F$ follows from the algebro-geometric theory of $G$ and the properties of the
Frobenius endomorphism. Similarly, most of the properties of Suziki groups
follows from the algebro-geometric theory of $G\_2$ together with the properties
of $\varphi$. As $\varphi$ is very similar to the Frobenius endomorphism this
works almost the same way as if $\varphi$ were indeed a Frobenius endomorphism.


**Addendum**: As one simple example of the similarity of $\varphi$ to a Frobenius consider the problem of computing the order of the Suzuki groups. As the square of $\varphi$ is the Frobenius, the action of it on the tangent space at any fixed point is nilpotent. This implies that such a fixed point appear with multiplicity one in the Lefschetz fixed point formula and the order of its group of fixed points is thus equal to the Lefschetz trace on (étale) cohomology of the algebraic group $G\_2$. That cohomology can be canonically expressed in terms the action of the Weyl group on the character group of the maximal torus (see for instance example in SGA 4 1/2) and how $\varphi$ acts on that character group is essentially part of the definition of $\varphi$.


