Twisted forms of classical reductive group schemes are described in a unified way. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, called the augmented odd form algebras, consist of 2-step nilpotent groups with an action of the underlying commutative ring, hence the basic descent theory for them will be developed. Finally, classical isotropic reductive groups are described as odd unitary groups up to an isogeny.