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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.

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    $\begingroup$ It is unclear what you mean by a parameterization. Do you mean that it is diffeomorphic to ${\mathbb C}^n$ for some $n$? (I assume you know what "diffeomorphic" means.) Or an open subset of ${\mathbb C}^n$? For instance, does $SL(2,C)$ admit a parameterization in your sense? The notion "full descriptive parametric equation" is not well-defined. $\endgroup$ – Moishe Kohan Apr 9 at 1:06

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