Parabolic subgroups of an affine algebraic group

Let $G$ be a smooth affine algebraic group of finite type over $k$ (Group schemes of finite type over a field, Smooth morphism of schemes, Affine schemes and their coordinate rings). A closed subgroup scheme $P\subseteq G$ is a parabolic subgroup if the fppf quotient $G\mathrm{/}P$ is representable and proper over $k$; when $G\mathrm{/}P$ is an integral $k$-variety, this is equivalent to completeness (Homogeneous spaces of smooth affine groups are separated schemes, Proper morphisms, Complete varieties). The unipotent radical $R_u(P)\subseteq P$ is the largest smooth connected normal unipotent subgroup (Unipotent algebraic groups and unipotent representations), the radical $R(P)$ is the largest smooth connected normal solvable subgroup (The derived subgroup, the derived series and solvable algebraic groups, Radical, unipotent radical, semisimple and reductive algebraic groups), and a Levi subgroup of $P$ is a smooth closed subgroup $L\subseteq P$ such that the multiplication map $L⋉R_u(P)\to P$ is an isomorphism. Borel subgroups and the equivalence 'parabolic iff, over $k^{\mathrm{a}}$, contains a Borel subgroup' are treated on this page in Solvable subgroups, the radical, and the Borel intersection ↗ and Borel subgroups, maximal tori and Borel pairs.

The conditional proper-quotient definition above uses no choice principle. Assume the Axiom of Choice for the following supplemental representability and structural assertions (The Axiom of Choice). Representability of $G\mathrm{/}P$ is a theorem for smooth affine $G$ and closed $P$ (Homogeneous spaces of smooth affine groups are separated schemes); For an integral representable quotient, properness is equivalent to completeness in the convention of Complete varieties. For a general quotient, properness is the defining condition; no integrality is assumed. The definition does not assume $P$ smooth or connected; in a smooth connected affine group such a parabolic is connected and self-normalizing by Solvable subgroups, the radical, and the Borel intersection ↗. Smooth parabolic subgroup varieties form the class to which the standard $P_I$ and Levi classification below applies; nonsmooth Frobenius-thickened proper-quotient subgroups are retained by this general definition. The definition makes no reference to split reductive structure, so that the standard-parabolic theory can establish the equivalence for smooth subgroup varieties. In the Levi decomposition below the subgroup $L$ is required only to be smooth and closed; when $P$ is smooth, its unipotent radical is smooth and the product decomposition is a statement about schemes.