If I have Lie group and don't have its product law but know its Lie algebra then could I reproduce the product law globally? Or at least in the vicinity of identity?
More precisely if I consider two specific elements of the group, say $A=e^{a^iE_i}$ and $B=e^{b^iE_i}$ where $E_i$ s are a basis for the Lie algebra and I know $a_i$ and $b_i$ as given constants then could I compute $c_i$ where
$$AB=C=e^{c^iE_i}$$ from the Lie algebra structure constants?
Yes, if the group is simply connected. In particular any two Lie groups with the same Lie algebra have the same universal cover, so are isomorphic in a neighborhood of the identity.