sl2 triple

Remarks

Example: in $\mathfrak{sl}_2(k)$, take $e = E_{12}$, $h = E_{11} - E_{22}$ and $f = E_{21}$. Then $[e, f] = E_{11} - E_{22} = h \ne 0$. Also $he = E_{12}$ and $eh = -E_{12}$, so $[h, e] = 2e$; similarly $[h, f] = -2f$.

Non-example: $(h, e, 0)$ with $h \ne 0$ is never an $\mathfrak{sl}_2$-triple, since then $[e, f] = 0 \ne h$, so clause 2 fails.

If $(h, e, f)$ is an $\mathfrak{sl}_2$-triple, the linear map $\mathfrak{sl}_2(k) \to \mathfrak g$ sending the standard basis to $(h, e, f)$ is a Lie algebra homomorphism. The element $e$ is called the nilpositive element and $f$ the nilnegative element of the triple. Some authors list the triple as $(e, h, f)$.