Exercise 9, § 5

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. Show that conditions (i), (ii), (iii) of Exerc. 16 of §3 are equivalent to:

(v) $\mathfrak{g}$ is decomposable and has the same rank as $\mathfrak{g}/\mathfrak{n}_V(\mathfrak{g})$.

(If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, the condition “$\mathfrak{g}$ has the same rank as $\mathfrak{g}/\mathfrak{n}_V(\mathfrak{g}$)” is equivalent to saying that $\mathfrak{h}\cap \mathfrak{n}_V(\mathfrak{g}) = 0$, i.e. that $\mathfrak{h}$ has no nilpotent element $\not= 0$ (cf. §2, Exerc. 14). Deduce the equivalence of (i) and (v).)