Forms of the Classical Lie Algebras
The discussion in this chapter may be regarded as motivated by the problem of determining all Lie algebras L with non-singular Killing form over an arbitrary field F (characteristic ≠ 2, 3). By Chapter I, § 7, the problem reduces to the case where 1D50F is simple. If ℨ is the centroid of L, then L is normal simple when regarded as a Lie algebra over ℨ moreover, if S is a ℨ-linear transformation of L, we have TrF (S) = Tℨ/F (Trℨ (S)), where Tℨ/F is the trace in the field extension ℨ/F (cf. [223, p. 66]). It follows that L has non-singular Killing form over ℨ, and that T ℨ/F is not zero, hence that ℨ/F is a separable extension. Thus the problem is reduced to the study of finite separable extensions of F, and to the case where L is normal simple. Assuming L normal simple over F, let Ω be an algebraically closed extension of F. Then L Ω has non-singular Killing form, and so is a classical simple Lie algebra over Ω therefore L Ω belongs to one of an already determined set of isomorphism classes. The “problem of forms” is that of describing L when L Ω is known, i.e., of determining the F-isomorphism classes of algebras L such that L Ω belongs to a given Ω-isomorphism class.
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