Let $\mathfrak{g}$ be a finite dimensionsl Lie algebra over $\mathbb{C}$. Denote its derivation algebra and automorphism group by $Der(\mathfrak{g})$ and $Aut(\mathfrak{g})$ respectively. Let $exp$ be the matrix exponential, i.e. $$exp(X)=\sum_{i=0}^{\infty}\dfrac{X^i}{i!}.$$
In general, we do not have $exp(Der(\mathfrak{g}))=Aut(\mathfrak{g})$. So I want to know under what conditions of $\mathfrak{g}$ can make the above formula come true. Thanks in advance.