Let ($\pi , V$) be a finite dimensional representation of $SL_{2}$. Also, let $\alpha$ be highest weight vector. I want to show that for any $m>0$, then the following holds:
$\pi$($X$)($\pi$($Y$))$^{m}v=m$($\alpha - m +1$)($\pi$($Y$))$^{m-1}v$
How would presumably proof by induction work in practice?