Pseudo-reductive group

In mathematics, a pseudo-reductive group or k-reductive group over a field k is a smooth connected affine algebraic group defined over k whose unipotent k-radical is trivial. The unipotent k-radical is the largest smooth connected unipotent normal subgroup defined over k. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A k-reductive group need not be a reductive k-group (a reductive group defined over k). Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants).
Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) builds on Tits' work to develop a general structure...