Differential equation and Lie algebra

Question

If I have this differential equation: $$ \frac{d\vec{x}}{dt} = F(\vec{x}) $$ and when $F = A$ is a matrix we can have the solution: $$ \vec{x}(t) = e^{At} \; \vec{x} $$ But what if $F = \mathfrak{g} $ is a Lie algebra, then can we have: $$ \vec{x}(t) = e^{\mathfrak{g} t} \; \vec{x} \quad ?$$ and if that is true, then the Lie group $ G = e^\mathfrak{g} $ leads to this solution: $$ \vec{x}(t) = G \; e^t \; \vec{x} $$ Is this correct?