# The automorphism group of a Lie algebra

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).

• 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. – Moishe Kohan Apr 9 at 1:06