Exercise 4, § 2

Let $\mathfrak{g}$ be a Lie algebra with a basis $(x, y, z)$ such that $[x, y] =y, [x, z] =z$, $[y, z] = 0$. Let $\mathfrak{a}$ be the ideal $ky+kz$ of $\mathfrak{g}$. Then rk($\mathfrak{a}$) $= 2$ and rk($\mathfrak{g}$) $= 1$.