= 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 $A=B=mI_2$, giving
$$
\boxed{u_{3,+}(\mathbf0)=\sqrt m\begin{pmatrix}1\\0\\1\\0\end{pmatrix}.}
$$
For a massless particle moving along the positive third axis, $p=(E,0,0,E)$ with $E>0$, and
$$
A=\begin{pmatrix}0&0\\0&2E\end{pmatrix},\qquad B=\begin{pmatrix}2E&0\\0&0\end{pmatrix}.
$$
Keeping the same two-spinor label gives
$$
\boxed{u_{3,+}(E\widehat{\mathbf e}_3)=\sqrt{2E}\begin{pmatrix}0\\0\\1\\0\end{pmatrix}.}
$$
It has positive <helicity> $+1/2$ and positive <chirality> in the <Weyl representation>.
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