= Solution
Choose the <chirality> operator
$$
\Gamma=\gamma^0\gamma^1=\sigma_3,\qquad \Gamma^2=I,\qquad \{\Gamma,\gamma^\mu\}=0.
$$
A <Weyl spinor> is a <Dirac spinor> restricted to one <eigenspace> of $\Gamma$. The <chiral projectors> are $P_R=(I+\Gamma)/2$ and $P_L=(I-\Gamma)/2$. Consequently
$$
\boxed{\psi_R=P_R\psi=\begin{pmatrix}\psi_+\\0\end{pmatrix},\qquad
\psi_L=P_L\psi=\begin{pmatrix}0\\\psi_-\end{pmatrix}.}
$$
The two independent <Weyl spinors> each contain one complex component. Since $[S(\theta),\Gamma]=0$, each <chirality> is preserved by every connected <Lorentz transformation>, with $\psi_R\mapsto e^{\theta/2}\psi_R$ and $\psi_L\mapsto e^{-\theta/2}\psi_L$. A spatial reflection exchanges the two <chiralities>.
The left/right names depend on the sign convention for $\Gamma$. Choosing $\gamma_0\gamma_1=-\sigma_3$ as the <chirality> operator exchanges the labels $L$ and $R$; the two invariant one-component <representations> are the same.
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