Example 2

Let $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$. If $x=(\gamma -\alpha )\in \mathfrak{g}$, an easy calculation gives

det(T $-$ ad $x$) $= T^3-4(\alpha \beta +\gamma^2)T$.

If the characteristic of $k$ is $\not= 2$, then rk($\mathfrak{g}$) $= 1$ and the regular elements are those $x$ such that $\alpha \beta +\gamma^2\not= 0$.