I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak g}\to U_{L}$ - homomorphism of their universal enveloping algebras (i.e. $\mathfrak g$ can be considered as a Lie subalgebra of $U_L$).
By trivial examples I mean when $\mathfrak g\subset\ker\psi$ or there exists a homomorphism $\phi\colon\mathfrak g\to L$.
Even more interesting question is about any examples of when:
1) $L$ is simple or a direct sum of simple algebras and
2) there is a homomorphism $\alpha\colon L\to\mathfrak{gl(g)}$ (i.e. $\mathfrak g$ is an $L$-module)
I guess the first question had already been discussed somewhere (or will in the nearest future) since it was interesting to find Lie subalgebras of Weyl algebra: http://arxiv.org/abs/math/0504224 But I don't know any reference since googling such a question is a naive way to search this (or I consider it so).
