# Constructive proof that the Lie functor is faithful?

I am wondering how to show that the Lie functor taking Lie groups to Lie algebras is faithful? In particular, I am looking for a constructive proof, since I am working in the context of synthetic differential geometry (and thus restricted to intuitionistic logic).

Sorry if this is an obvious question, but my algebra background isn't that strong, and I mostly taught myself what little Lie theory I know.

• This is false unless you restrict to connected Lie groups. I am not sure what's constructive, but what proofs have you seen? There's two that I know: one that first passes through formal group laws (in Serre's book on Lie groups) and one that passes through foliations (you can find this in Lee's book). – user98602 Jan 4 '16 at 14:34

For non-connected Lie groups the Lie functor is not faithful, see page $35$ here for "explicit counterexamples".
• Yes, there are many explicit counterexamples. Just consider any two non-connected Lie groups $G$ and $H$ such that their respective identity connected components $G_I$ and $H_I$ are Lie groups themselves - see my edit. – Dietrich Burde Jan 5 '16 at 11:10
• To be rather silly: $G = \Bbb Z/2$, $H = *$. – user98602 Jan 6 '16 at 18:42