TITLE: Suzuki and Ree groups, from the algebraic group standpoint
QUESTION [22 upvotes]: 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?

REPLY [7 votes]: The second part of Karsten Naert's thesis constructs Suzuki—Ree groups as groups of points of "algebraic groups over $\mathbb{F}_{\!\sqrt{p}}$". Such fields being nonexistent, these "algebraic groups" are understood not as schemes, but as the so-called "twisted schemes". They live in the category of pairs $(X, \Phi_X)$, where $X$ is a scheme and $\Phi_X\in \operatorname{End}(X)$ satisfies $\Phi_X\circ\Phi_X=F_X$, the Frobenius morphism.
Another way to describe Suzuki—Ree groups as something "defined by equations" (not algebraic, but difference-algebraic) is to compare two fundamental representations of the ambient Chevalley group in characteristic 2 or 3, whose highest weights $\lambda$ and $\mu$ are interchanged by the exceptional symmetry. These representations happen to have the same dimension, so one considers the group of all $g$ such that $\tau g_
\lambda=g_\mu\tau$, where $\tau^2$ is the Frobenius morphism. This produces the Suzuki—Ree groups, requires no generators and relations and works over any ring of characteristic $p$ admitting such $\tau$ (in particular, the field of definition need not be perfect).