Ado’s theorem is equivalent with the following:
Theorem. Let be a local Lie group. Then there is a real, finite dimensional vector space and an injective, local group morphism from (a neighbourhood of the neutral element of) to , the linear group of $V$.
Any proof I am aware of, (see this post for one proof and relevant links), mixes the following ingredients:
– the Lie bracket and the BCH formula,
– either reduction to the nilpotent case or (nonexclusive) use of differential equations,
– the universal enveloping algebra.
WARNING: further I shall not mention the “local” word, in the realm of spaces with dilations everything is local.
– locally compact groups with dilations instead of Lie groups
– locally compact conical groups instead of vector spaces
– linearity in the sense of dilation structures instead of usual linearity.
Conjecture: For any locally compact group with dilations there is a locally compact conical group and an injective morphism such that for every the map is differentiable.
In this frame:
– we don’t have the corresponding Lie bracket and BCH formula, see the related problem of the noncommutative BCH formula,
– what nilpotent means is no longer clear (or needed?)
– we don’t have a clear tensor product, therefore we don’t have a correspondent of the universal enveloping algebra.
Nevertheless I think the conjecture is true and actually much easier to prove than Ado’s theorem, because of the weakening of the conclusion.