Let $M$ be a smooth manifold. Then there is a Lie algebra structure on $C^{\infty}(M)$ (considered as a vector space).
I want to find the corresponding Lie group, $G$. I'm interested in how this works in general, so I will not specify a specific Lie bracket.
I know that there is a correspondence between Lie groups and Lie algebras given by the exponential map.
$$\exp: C^{\infty}(M)\to G$$ For each $f\in C^{\infty}(M)$, $\exp(f):=\gamma(1)$ where $\gamma:\mathbb{R}\to G$ is the unique $1$-PSG of $G$ (that is, continuous homomorphism of groups) such that $\dot{\gamma}(0)=f$.
How can I use this to find $G$?
Obviously, $G$ will depend on the Lie bracket, but I don't see how to use this. Do I need to find a basis for $C^{\infty}(M)$ and try to work out how $\exp$ acts on this basis? I don't really see how this could be done, since nothing is known about $G$.