Lemma 4

Let $(x, h, y)$ and $(x, h', y')$ be $\mathfrak{s}\mathfrak{l}_2$-triplets in $\mathfrak{g}$. There exists $z\in \mathfrak{g}$ such that ad$_{\mathfrak{g}}z$ is nilpotent and such that

Let $\mathfrak{n}=$ Ker(ad $x$)$\cap$ Im(ad $x$). For all $p\in \mathbf{Z}$, let $\mathfrak{g}_p=$ Ker(ad $h-p$). By §1, no. 3 (applied to the adjoint representation of $kx+ky+kh$ on $\mathfrak{g}$), we have that $\mathfrak{n}\subset \sum_{p>0}\mathfrak{g}_p$, so ad$_{\mathfrak{g}}n$ is nilpotent for all $n\in \mathfrak{n}$, and $[h,\mathfrak{n}] =\mathfrak{n}$. We have $[x, h'-h] = 0$ and $[x, y-y'] =h'-h$, so $h'-h\in \mathfrak{n}$. By Lemmas 2 and 3, there exists $z\in \mathfrak{n}$ such that $e^{ad_{\mathfrak{g}}z}h=h'$. Since $z\in$ Ker ad$_{\mathfrak{g}}x$, we have $e^{ad_{\mathfrak{g}}z}x=x$. Lemma 1 now proves that $e^{ad_{\mathfrak{g}}z}y=y'$. Q.E.D.

Let G be a group of automorphisms of $\mathfrak{g}$. Then two $\mathfrak{s}\mathfrak{l}_2$-triplets $(x, h, y)$, $(x', h', y')$ are said to be G-conjugate if there exists $g\in G$ such that $gx=x'$, $gh=h', gy=y'$.