Since $\gamma_1^2=\gamma_2^2=-\Bbb I_4$, $\frac{\gamma_1^2-\gamma_2^2}{2}=\Bbb O_4$.

Hint: verify $\gamma_1^2=\gamma_2^2=-\Bbb I_4$. (You may wish to also check $\{\gamma_1,\,\gamma_2\}=\Bbb O_4$, so $\alpha^2=\alpha^{\dagger2}=0$.)