Solvability of the radical of the Killing form
= Solvability of the radical of the Killing form
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The radical $\mathfrak g^\perp=\{x:\kappa(x,\mathfrak g)=0\}$ of the Killing form is a solvable ideal. Invariance makes it an ideal, and the <Cartan solvability criterion> applied to its adjoint image proves solvability.