Automorphisms of the Classical Algebras
The automorphism groups of classical Lie algebras, in the sense of the previous chapter, have been studied for the four “great classes” A-D by Jacobson , considering the most natural realizations of these algebras. A unified approach has been made by the author , substituting certain combinations of algebraic operations for the exponential functions used in the fundamental work of Gantmacher  in the complex case. Where the author’s results are incomplete (in case the ground field is not algebraically closed), they have been completed by Steinberg . Indeed, Steinberg is able to deal with characteristics 2 and 3 as well, since he obtains his Lie algebras by Chevalley’s process (Chap. II, § 3) from a complex semisimple Lie algebra. We reproduce here the results of Steinberg, restricted to the case of classical algebras in our sense, as well as giving essentially Chevalley’s results on the general structure of the groups of Chevalley, when regarded as subgroups of the automorphism groups of classical algebras. Finally, we give interpretations for these results in terms of the natural realizations for types A-D, as well as for the exceptional algebras.
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