Ado-Iwasawa extras

Let L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation p of L. Let S be a subnormal subalgebra of L, let F be a saturated formation of soluble Lie algebras and suppose that S is an element of a. I show that there exists a module V with the extra property that it is F-hypercentral as S-module. Further, there exists a module V which has this extra property simultaneously for every such S and 3, along with the Hochschild extra that rho(x) is nilpotent for every x is an element of L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.