Let $U( g)$ be the enveloping algebra of a semi-simple Lie algebra $ g$. Very little is known about the nature of $Aut\, U( g)$. However, if $G$ is a finite subgroup of $Aut\, U( g)$ then very general results of Lorenz-Passman and of Montgomery can be used to relate $Spec\, U( g)$ to $Spec\, U( g)^G$. As noted by Alev-Polo one may read off the Dynkin diagram of $ g$ from $Spec\, U( g)$ and they used this to show that $U( g)^G$ could not be again the enveloping algebra of a semi-simple Lie algebra unless $G$ is trivial. Again let $U$ be the minimal primitive quotient of $U( g)$ admitting the trivial representation of $ g$. A theorem of Polo asserts that if $U^G$ is isomorphic to a similarly defined quotient of $U( g'): g'$ semi-simple, then $ g\cong g'$. However in this case one cannot say that $G$ is trivial. The main content of this paper is the possible generalization of Polo's theorem to other minimal primitive quotients. A very significant technical difficulty arises from the Goldie rank of the almost minimal primitive quotients being $>1$. Even under relatively strong hypotheses (regularity and integrality of the central character) one is only able to say that the Coxeter diagrams of $ g$ and $ g'$ coincide. The main thrust of the proofs is a systematic use of the Lorenz-Passman-Montgomery theory and the known very detailed description of $Prim\, U( g)$. Unfortunately there is a severe lack of good examples. During this work some purely ring theoretic results involving Goldie rank comparisons and skew-field extensions are presented. A new inequality for Gelfand-Kirillov dimension is obtained and this leads to an interesting question involving a possible application of the intersection theorem.