Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra. It is well-known that the the automorphism group of $\mathfrak{g}$, $\operatorname{Aut}(\mathfrak{g})$, is an algebraic group.
How should $\mathfrak{g}$ be so that $\operatorname{Aut}(\mathfrak{g})$ admits a parametrization (i.e. to give full descriptive parametric equations of $\operatorname{Aut}(\mathfrak{g})$)?
For instance, if $\mathfrak{g}$ is a $3$-dimensional complex solvable Lie algebra, then $\operatorname{Aut}(\mathfrak{g})$ admits a paremetrization (see Biggs, Remsing: "Invariant control systems on Lie groups". Lie Groups, Differential Equations, and Geometry: Advances and Surveys).
Thanks in advance.