 Principal sl2 subalgebra

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.

The infinitesimal embedding is a subalgebra of the semisimple Lie algebra; integrating it to a Lie-group homomorphism is a separate group-level construction.