The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a suitable 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 suitable 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?

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?