The principal nilpotent Borel orbit by height elimination

On a principal ${\mathfrak{s}\mathfrak{l}}_2$ string with highest weight $2m>0$, $\mathrm{ad}h$ has weights $-2m,-2m+2,\dots ,2m$. For every $j\ge 1$, if weight $2j$ occurs then weight $2j-2$ occurs, and the raising operator $\mathrm{ad}e$ maps the latter one-dimensional weight space onto the former. Summing over the strings in [F1] gives $[e,{\mathfrak{n}}_j^{+}]={\mathfrak{n}}_{j+1}^{+}(j\ge 1),$ with zero spaces understood above the maximal root height.

Let $x\in {\mathfrak{n}}^{+}$ have nonzero simple coefficients. The map $T\to ({\mathbb{C}}^{\times }{)}^r$, $t\mapsto ({\alpha }_i(t){)}_i$, is an isogeny: its character-lattice map has image the full-rank root lattice. An isogeny of complex tori is surjective on complex points, so conjugation by one $t\in T$ sets all simple coefficients of $x$ to $1$. Now suppose heights below $j+1$ already agree with those of $e$. Their height-$(j+1)$ difference lies in ${\mathfrak{n}}_{j+1}^{+}$; by [step 1.1] choose $u_j\in {\mathfrak{n}}_j^{+}$ whose bracket with $e$ cancels that difference. The finite exponential $\exp (\mathrm{ad}{u}_j)$ changes no height at most $j$ and changes height $j+1$ by $[u_j,e]$; all its higher commutators have height at least $j+2$. Induction over the finitely many root heights conjugates $x$ to $e$ by $U$.

Conversely, conjugation by $T$ scales each simple-root coefficient by a nonzero scalar. For $u\in U$, every commutator in $\exp (\mathrm{ad}u)e-e$ has height at least two, because $u$ has positive height and $e$ has height one. Thus $U$ fixes the simple-root coefficients, and $B=TU$ preserves their nonvanishing. The inclusion in the right side and the reverse inclusion from step 2.1 prove the equality. The rank-zero case has ${\mathfrak{n}}^{+}=0$, $e=0$ and both sides are the singleton $\{0\}$.