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Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.

Does there necessarily exist a Levi subalgebra of $\mathfrak g$ which is $G$-invariant?

By Levi subalgebra I mean a semisimple complement of the solvable radical, as in the Levi-Malcev theorem. My field is $\mathbb Q$... but if needed I could probably deal with extensions of scalars.

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# Equivariant Levi subalgebras.

Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.

Does there necessarily exist a Levi subalgebra of $\mathfrak g$ which is $G$-invariant?