A Subdirect-union Representation for Completely Distributive Complete Lattices

4. 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 representation x—*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 = 3Ï as above.