Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ for all t.
Now, I only know how to construt this for matrices (or bounded operators if you wish) as it coincides in this context with the standard exponential map. Despite, many proofs use this construction, so I am not sure if such a map exists for every Lie-algebra. In case that this is true, I would love to know the properties of this map.
In particular, I am interested in knowing more about the differentiability properties of such a map.