Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two definitions are equivalent: any abstract group acts as a collection of permutations of its underlying set, and this action is faithful.

Similarly, rings can be defined abstractly as sets with a pair of binary operations satisfying certain identities, or concretely as a collection of endomorphisms of an abelian group. There is a "Cayley's theorem" here as well: any abstract ring acts as a collection of endomorphisms of its underlying abelian group, and this action is faithful.

The situation for Lie algebras seems much less clear to me. The adjoint representation is not generally faithful, and Ado's theorem comes with qualifications and doesn't have the simplicity of the two theorems above. For me, the problem is that I don't have a good sense of what the concrete definition of a Lie algebra is supposed to be.

I suspect that a good concrete definition of a Lie algebra is as a space of derivations on some algebra closed under commutator. In that case, is it correct to say that a Lie algebra acts faithfully as derivations on its universal enveloping algebra? Is this a good analogue of Cayley's theorem?

(Motivation: in the books on Lie algebras I have read, the authors verify that Lie algebras which occur in nature satisfy alternativity and the Jacobi identity, but I have never seen any simple justification that these axioms are "enough" in the same way that Cayley's theorem tells you that the axioms for a group or a ring are "enough." There is just Ado's theorem which, again, comes with qualifications and is hard.)