Let $E$ be a finite banach space, and let the Lie-algebra $$E\oplus[E,E]\oplus[E,[E,E]]\oplus....$$ be the formal Lie-series $L$.
So the space of formal-Lie-series $L$ can be seen as the completion of the free Lie-algebra over $E$ $L_x$ and hence is finite dimensional. By Lie's third theorem, one can associate to $L$ a Lie-group $G$.
Now my question is, how does the dimension of $G$ relate to the dimension of $L$. Is there some known correspondence between the dimension between the both. Does $G$ even have to be finite dimensional?
Best wishes