Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.
Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.