Lemma 1

Let $x, h, y, y'\in \mathfrak{g}$. If $(x, h, y)$ and $(x, h, y')$ are $\mathfrak{s}\mathfrak{l}_2$-triplets in $\mathfrak{g}$, then $y=y'$.

Indeed, $y-y'\in$ Ker(ad$_{\mathfrak{g}}x$) and (ad$_{\mathfrak{g}}h$)$(y-y') =-2(y-y')$. But ad$_{\mathfrak{g}}x$ is injective on Ker($p+$ ad$_{\mathfrak{g}}h$) for every integer $p >0 ($§1, no. 2, Cor. of Prop. 2).