Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 3 / c / Solution

Put $h_0=\psi (h)$ and $e_0=\psi (e)={\sum }_{i=1}^{\ell }{e}_i$, where $0\ne e_i\in {\mathfrak{g}}_{{\alpha }_i}$. Since $[h_0,e_0]=2e_0$ and the root spaces are a direct sum, ${\alpha }_i(h_0)=2$ for every simple root. Every root has simple-root coefficients of one sign, so no root vanishes on $h_0$. The Regular element criterion in a Cartan subalgebra therefore shows that $h_0$ is regular.

Restrict the Adjoint representation of $\mathfrak{g}$ along $\psi$. By Complete reducibility of semisimple Lie algebra representations, it is a direct sum of finite-dimensional sl2 Lie algebra modules. The $h_0$-eigenvalues on a root space are twice the heights of the roots, so they are all even. Each irreducible summand consequently has even highest weight, contains exactly one zero-weight vector, and has a one-dimensional kernel for the raising operator $e_0$ by the Classification of finite-dimensional sl2 representations.

Because $h_0$ is regular, its zero-weight space in $\mathfrak{g}$ is precisely $\mathfrak{t}$ and has dimension $\ell$. There are therefore exactly $\ell$ irreducible summands, whenceThus $e_0$ is regular; equivalently it is a principal nilpotent element in the given Principal sl2 subalgebra.