Inner form
In mathematics, an inner form of an algebraic group  over a field  is another algebraic group  such that there exists an isomorphism  between  and  defined over  (this means that  is a -form of ) and in addition, for every Galois automorphism  the automorphism  is an inner automorphism of  (i.e. conjugation by an element of ).

Through the correspondence between -forms and the Galois cohomology  this means that  is associated to an element of the subset  where  is the subgroup of inner automorphisms of .

Being inner forms of each other is an equivalence relation on the set of -forms of a given algebraic group.

A form which is not inner is called an outer form. In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group  on the Dynkin diagram of  (induced by its action on , which preserves any torus and hence acts on the roots). Two groups are inner forms of each other if and only if the actions they define are the same.

For example, the -forms of  are itself and the unitary groups  and . The latter two are outer forms of , and they are inner forms of each other.

References

Algebraic groups