  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.
