# solvable lie algebras for which adjoint representation is direct sum of irreducible representations

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?