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Find all solvable lie algebras for which adjoint representation is decomposable into direct sum of irreducible representations.

My attempt:

Let $\mathfrak{g}$ such a Lie algebra.

$\mathfrak{g}$ solvable $\Rightarrow rad(\mathfrak{g}) = \mathfrak{g}$

Adjoint representation is decomposable into direct sum of irreducible representations $\Leftrightarrow \mathfrak{g}$ is reductive $\Leftrightarrow rad(\mathfrak{g}) = \mathfrak{z}(\mathfrak{g})$.

So necessary condition is that $\mathfrak{g}=\mathfrak{z}(\mathfrak{g})$, i.e. $\mathfrak{g}$ is abelian.

Reciprocally, if $\mathfrak{g}$ is abelian, it is solvable so $rad(\mathfrak{g}) = \mathfrak{g}$, but $\mathfrak{g}=\mathfrak{z}(\mathfrak{g})$ so $rad(\mathfrak{g}) = \mathfrak{z}(\mathfrak{g})$ and $\mathfrak{g}$ is reductive.

Is that a correct proof?

Thank you for your help.

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