Given a principal (regular) nilpotent element $e$ in the Lie algebra $\mathfrak g$ of a complex semisimple algebraic group $G$, let $\mathfrak s=(e,f,h)$ be an $\mathfrak{sl}_2$-triple for $e$. Then the stabilizer of $h$ in $G$ is a maximal torus $T$ acting on the $2$-eigenspace $\mathfrak g_2$ of $\operatorname{ad}h$ with an open orbit (orbit of $e$).
This must be one of the very basic examples of an affine (in fact, linear) toric variety, and it must be doubtlessly studied in much detail somewhere. What text would you recommend to read about it?
