A subdirect-union representation for completely distributive complete lattices

Abstract

Proof of Theorem 2. Harish-Chandra[l] and others have proved that every Lie algebra over a field of characteristic zero has a faithful representation.Consequently by Lemma 4, 8 has a faithful representationx-*Qx whose matrices have elements in an algebraic extension $ of g such that t(QxQv) =0 for all x of 31 and all y of 8. We now apply another form of Cartan's criterion for solvability which states that if t(Ai)=0 for all A in a Lie algebra 21 of linear transformations, than 21 is solvable, and deduce that the deal 93 of all x of 8 such that t(QxQy) =0 for every y of 8 is solvable.This proves the theorem for we now have 93 = 3 as above.