Exercise 9, § 2

Let $\mathfrak{g}$ be a Lie algebra with a basis $(x, y)$ such that $[x, y] =y$. Let $\mathfrak{k}=ky$, and $\varphi :\mathfrak{g}\rightarrow \mathfrak{g}/\mathfrak{k}$ the canonical morphism. The element 0 of $\mathfrak{g}/\mathfrak{k}$ is regular in $\mathfrak{g}/\mathfrak{k}$ but is not the image under $\varphi$ of a regular element of $\mathfrak{g}$.