Exercise 6, § 4

Let $g$ be a nilpotent Lie algebra and $a$ a commutative ideal of $g$. The following conditions are equivalent: (a) $a$ is a maximal commutative ideal of $g$; (b) $a$ is a maximal commutative subalgebra of $g$; (c) $a$ is equal to its centralizer $a'$ in $g$. (The implications not (a) $\Rightarrow$ not (b) $\Rightarrow$ not (c) are obvious. If $a' \neq a$, there exists by Proposition 1 an ideal $a''$ of $g$ such that $a \subset a'' \subset a'$ and $\dim a''/a = 1$. Then $a'' = a + Kx$, hence $[a'', a''] \subset [a, a] + [x, a] = \{0\}$ and hence (a) is false.)