Twisted forms of classical groups

Abstract

We give a unified description of twisted forms of classical reductive groups schemes. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, augmented odd form algebras, consist of $2$-step nilpotent groups with an action of the underlying commutative ring, hence we develop basic descent theory for them. In addition, we describe classical isotropic reductive groups as odd unitary groups up to an isogeny.