## Return to Answer

3 added 60 characters in body

It would help to place your question in the context of the foundational 1965 paper Groupes reductifs by Borel and Tits, freely available online from NUMDAM: http://archive.numdam.org/

For example, their Section 4 studies centralizers of maximal $k$-split tori in terms of roots, parabolic subgroups, Levi subgroups. This set-up was used by Tits to codify many details of the classification of semisimple groups over fields of special interest: finite, local, algebraic number fields, etc.
Relative to a field of definition, certain Levi subgroups of parabolic subgroups are natural examples of the centralizers you want. Your proposed example needs to be placed more carefully within this Borel-Tits framework, I think.

The story about structure and classification of reductive groups over arbitrary fields is a long one, but the Tits strategy is to start with the known split groups and then adapt the Dynkin diagram to a field of definition. See his paper in the proceedings of the 1965 AMS Summer Institute at Boulder, available freely online through AMS e-math in the first part of the volume: MR0224710 (37 #309) Tits, J. Classification of algebraic semisimple groups. 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 33--62 Amer. Math. Soc., Providence, R.I., 1966. See the Web page http://www.ams.org/online_bks/pspum9/

More details were worked out by a student of Tits at Bonn: see MR0432776 (55 #5759) Selbach, Martin Klassifikationstheorie halbeinfacher algebraischer Gruppen. Diplomarbeit, Univ. Bonn, Bonn, 1973. Bonner Mathematische Schriften, Nr. 83. Mathematisches Institut der Universität Bonn, Bonn, 1976. v+140 pp.

Your group is of inner type A in the classification, using the Dieudonne determinant notation. So this really isn't so "typical", but occurs in the Tits list. The "split" data in his diagrams is somewhat independent of the ground field, but the remaining classification problem for anisotropic groups depends strongly on the field.

2 added 1277 characters in body

The story about structure and classification of reductive groups over arbitraryfields is a long one, but the Tits strategy is to start with the known splitgroups and then adapt the Dynkin diagram to a field of definition. See hispaper in the proceedings of the 1965 AMS Summer Institute at Boulder, availablefreely online through AMS e-math in the first part of the volume:MR0224710 (37 #309) Tits, J.Classification of algebraic semisimple groups. 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 33--62 Amer. Math. Soc., Providence, R.I., 1966. More details were workedout by a student of Tits at Bonn: see MR0432776 (55 #5759) Selbach, MartinKlassifikationstheorie halbeinfacher algebraischer Gruppen. Diplomarbeit, Univ. Bonn, Bonn, 1973. Bonner Mathematische Schriften, Nr. 83. Mathematisches Institut der Universität Bonn, Bonn, 1976. v+140 pp.

Your group is of inner type A in the classification, using the Dieudonnedeterminant notation. So this really isn't so "typical", but occurs in theTits list. The "split" data in his diagrams is somewhat independent ofthe ground field, but the remaining classification problem for anisotropicgroups depends strongly on the field.

1

It would help to place your question in the context of the foundational 1965 paper Groupes reductifs by Borel and Tits, freely available online from NUMDAM: http://archive.numdam.org/

For example, their Section 4 studies centralizers of maximal $k$-split tori in terms of roots, parabolic subgroups, Levi subgroups. This set-up was used by Tits to codify many details of the classification of semisimple groups over fields of special interest: finite, local, algebraic number fields, etc.
Relative to a field of definition, certain Levi subgroups of parabolic subgroups are natural examples of the centralizers you want. Your proposed example needs to be placed more carefully within this Borel-Tits framework, I think.