Proposition 5

Let g be a Lie algebra and r its radical. (a) If ρ is a finite-dimensional representation of g and β is the associated bilinear form, r and $\mathscr{D}$g are orthogonal with respect to β. (b) r is the orthogonal of $\mathscr{D}$g with respect to the Killing form. Let x, y be in g, $z \in r$. Then $[y, z] \in \mathscr{D}g \cap r$ and hence

(Theorem 1). Hence (a). Let r' be the orthogonal of $\mathscr{D}$g with respect to the Killing form. It is an ideal of g (§ 3, no. 6, Proposition 7 (a)) which contains r by the above. On the other hand, the image s of r' under the adjoint representation of g is solvable (Theorem 2) and hence r' is solvable being a central extension of s. Hence r' ⊂ r.