 Solution

Both blocks of the given rotation matrix carry the same spin one-half representation, so a stationary spin-up state along the third axis uses the $+1$ eigenspinor ${\xi }_3=(1,0{)}^T$ of ${\sigma }_3$. At rest $A=B=m{I}_2$, givingFor a massless particle moving along the positive third axis, $p=(E,0,0,E)$ with $E>0$, andKeeping the same two-spinor label givesIt has positive helicity $+1/2$ and positive chirality in the Weyl representation.