= Solution
Invariance of the <Killing form> gives
$$
\kappa([a,x],y)=\kappa(x,[y,a]).
$$
If $x\in\mathfrak g^\perp$, the right-hand side vanishes for every $y$, so $[a,x]\in\mathfrak g^\perp$. Thus $\mathfrak g^\perp$ is an ideal.
Let $\mathfrak h=\operatorname{ad}(\mathfrak g^\perp)\subseteq\mathfrak{gl}(\mathfrak g)$. For $x\in[\mathfrak g^\perp,\mathfrak g^\perp]$ and $y\in\mathfrak g^\perp$,
$$
\operatorname{tr}(\operatorname{ad}_x\operatorname{ad}_y)
=\kappa(x,y)=0.
$$
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.
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