Exercise 4, § 11

Let $V$ be a vector space of dimension $n\geqslant2$, $\mathfrak g=\mathfrak{sl}(V)$ and $G=\mathrm{PGL}(V)$, identified with a group of automorphisms of $\mathfrak g$. An $\mathfrak{sl}_2$-triplet in $\mathfrak g$ gives $V$ a faithful $\mathfrak{sl}(2,k)$-module structure, and conversely any such structure arises from an $\mathfrak{sl}_2$-triplet; an $\mathfrak{sl}_2$-triplet is principal if and only if the corresponding $\mathfrak{sl}(2,k)$-module is simple; two $\mathfrak{sl}_2$-triplets are G-conjugate if and only if the corresponding $\mathfrak{sl}(2,k)$-modules are isomorphic. Deduce that the G-conjugacy classes of $\mathfrak{sl}_2$-triplets correspond bijectively with the families $(m_1,m_2,\ldots)$ of integers $\geqslant0$ such that