Reducibility of an sl2 Verma module ($M_{\lambda }$ reducible iff $\lambda \in {\mathbb{Z}}_{\ge 0}$)

The Verma module $M_{\lambda }$ has basis $(f^r{v}_{\lambda }{)}_{r\ge 0}$ andIt is reducible exactly when $\lambda =m\in {\mathbb{Z}}_{\ge 0}$; then its unique proper nonzero submodule is generated by $f^{m+1}{v}_{\lambda }$ and is isomorphic to $M_{-m-2}$.