sl2 subalgebra associated with a root
= sl2 subalgebra associated with a root
{c}
{title2=$\mathfrak m_\alpha\cong\mathfrak{sl}_2$}
For every root $\alpha$ of a complex semisimple Lie algebra, there are root vectors $e_\alpha\in\mathfrak g_\alpha$ and $f_\alpha\in\mathfrak g_{-\alpha}$ and an element $h_\alpha\in\mathfrak t$ satisfying
$$
[h_\alpha,e_\alpha]=2e_\alpha,qquad
[h_\alpha,f_\alpha]=-2f_\alpha,qquad
[e_\alpha,f_\alpha]=h_\alpha.
$$
Their span is an $\mathfrak{sl}_2$ subalgebra.