Definition 1

An $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$ is a sequence $(x, h, y)$ of elements of $\mathfrak{g}$, distinct from $(0,0,0)$, such that

Let $(x, h, y)$ be an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$. The linear map $\tau$ from $\mathfrak{s}\mathfrak{l}(2, k)$ to $\mathfrak{g}$ such that $\tau (X_+) =x, \tau (H) =h, \tau (X_-) =y$ is a homomorphism which is non-zero and hence injective (since $\mathfrak{s}\mathfrak{l}(2, k)$ is simple), and with image $kx+kh+ky$. We thus obtain a canonical bijection from the set of $\mathfrak{s}\mathfrak{l}_2$-triplets in $\mathfrak{g}$ to the set of injective homomorphisms from $\mathfrak{s}\mathfrak{l}(2, k)$ to $\mathfrak{g}$. If $\mathfrak{g}$ is semi-simple and if $(x, h, y)$ is an $\mathfrak{s}\mathfrak{l}_2$-triplet in $\mathfrak{g}$, then $x$ and $y$ are nilpotent elements of $\mathfrak{g}$ and $h$ is a semi-simple element of $\mathfrak{g}$ (Chap. I, §6, no. 3, Prop. 4).