Proposition 7

Assume that $\mathfrak{g}$ is semi-simple. Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$. The following conditions are equivalent:

(i) $x$ is principal;

(ii) $h$ is principal;

(iii) $(x, h, y)$ is principal.

For $p\in \mathbf{Z}$, let $\mathfrak{g}^p=$ Ker(ad $h-p$). Let $\mathfrak{g}'=\sum_{p\in\mathbf{Z}}\mathfrak{g}^{2p}$. If $\mathfrak{g}$ is considered as

a module over $\mathfrak{a}=kx+kh+ky,\mathfrak{g}'$ is the sum of the simple submodules of odd dimension (§1, no. 2, Cor. of Prop. 2). Let $l$ (resp. $l'$) be the length of $\mathfrak{g}$ (resp. $\mathfrak{g}'$) considered as an $\mathfrak{a}$-module. By §1, no. 2,

dim(Ker ad $x$) $=l\geq l'=$ dim(Ker ad $h$)$\geq$ rk($\mathfrak{g}$).

The equivalence of (i) and (iii) follows immediately. On the other hand, condition (ii) means that dim(Ker ad $h$) $=$ rk($\mathfrak{g}$) and $\mathfrak{g}'=\mathfrak{g}$, in other words that

dim(Ker ad $h$) $=$ rk($\mathfrak{g}$) and $l=l'$. The equivalence of (ii) and the other conditions follows.