Principal sl2 subalgebra (${\mathfrak{sl}}_2^{\mathrm{pr}}$)

A principal sl2 subalgebra of a semisimple Lie algebra has semisimple generator $h_{\mathrm{pr}}=2{\rho }^{\vee }$. Restriction to it packages a representation's weights into ordinary ${\mathfrak{sl}}_2$ weight strings.