Exercise 10, § 2

Assume that $k$ is infinite. Let $\mathfrak{g}$ be a Lie algebra. Prove the equivalence of the following properties:

(i) rk($\mathfrak{g}$) $=$ dim($\mathfrak{g}$).

(ii) $\mathfrak{g}$ is nilpotent.

(iii) $\mathfrak{g}$ has only finitely-many Cartan subalgebras of dimension rk($\mathfrak{g}$).

(iv) $\mathfrak{g}$ has only one Cartan subalgebra.