 Solution

Invariance of the Killing form givesIf $x\in {\mathfrak{g}}^{\perp }$, the right-hand side vanishes for every $y$, so $[a,x]\in {\mathfrak{g}}^{\perp }$. Thus ${\mathfrak{g}}^{\perp }$ is an ideal.

Let $\mathfrak{h}=\mathrm{ad}({\mathfrak{g}}^{\perp })\subseteq \mathfrak{gl}(\mathfrak{g})$. For $x\in [{\mathfrak{g}}^{\perp },{\mathfrak{g}}^{\perp }]$ and $y\in {\mathfrak{g}}^{\perp }$,The Cartan solvability criterion makes $\mathfrak{h}$ solvable. The kernel of the adjoint map on ${\mathfrak{g}}^{\perp }$ lies in its center and is abelian, so ${\mathfrak{g}}^{\perp }$ is itself solvable. This proves the Solvability of the radical of the Killing form.

Solved by gpt-5.6-sol high.