= 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 \b[$h_0$ 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, whence
$$
\dim\ker(\operatorname{ad}e_0)=\ell.
$$
Thus \b[$e_0$ is regular]; equivalently it is a <principal nilpotent element> in the given <Principal sl2 subalgebra>.
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