Exercise 12, § 2

Assume that $k$ is finite. Construct an example of a Lie algebra $\mathfrak{g}$ having a Cartan subalgebra $\mathfrak{h}$ in which there exists no element $x$ such that $\mathfrak{h}=\mathfrak{g}^0(x)$. (Use the preceding exercise, and take $P =\mathfrak{h}^*.$)