Exercise 7, § 3

Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ a subalgebra of $\mathfrak{g}$ reductive in $\mathfrak{g}$. Let $\mathfrak{a}$ be a Cartan subalgebra of $\mathfrak{g}'$. Show that there exists a Cartan subalgebra of $\mathfrak{g}$ which contains $\mathfrak{a}$ (use Prop. 10 of §2). Deduce that rk($\mathfrak{g}'$)$\leq$ rk($\mathfrak{g}$) and that equality holds if and only if $\mathfrak{g}'$ has properties (i), (ii), (iii) of Prop. 3.