Exercise 3, § 11

Let $\mathfrak r$ be the radical of $\mathfrak g$, and $\mathfrak s=\mathfrak g/\mathfrak r$. Show the equivalence of:

(i) $\mathfrak g$ contains no $\mathfrak{sl}_2$-triplet;

(ii) $\mathfrak s$ contains no $\mathfrak{sl}_2$-triplet;

(iii) $\mathfrak s$ is anisotropic (§10, Exerc. 6);

(iv) $\mathfrak s$ contains no non-zero diagonalizable element (§3, Exerc. 10).

(Use Prop. 2 and Exerc. 10 a) of §3.)