= Solution
The ordering of <Weyl spinor>[Weyl components] depends on the supplied basis transformation. In this paper use $u_W=Uu_D=(\xi,\eta)^{\mathsf T}$. Applying the printed $U$ to the printed Dirac matrices gives
$$
\gamma_W^0=\begin{pmatrix}0&I\\I&0\end{pmatrix},\qquad
\gamma_W^i=\begin{pmatrix}0&-\sigma_i\\\sigma_i&0\end{pmatrix},\qquad
\gamma_W^5=\begin{pmatrix}I&0\\0&-I\end{pmatrix}.
$$
Thus $\xi$ is right chiral and $\eta$ left chiral in this ordering. The free <massless particle>[massless] <Weyl equations> are
$$
\boxed{i(\partial_t+\boldsymbol\sigma\cdot\boldsymbol\nabla)\xi=0,
\qquad i(\partial_t-\boldsymbol\sigma\cdot\boldsymbol\nabla)\eta=0.}
$$
They describe <spin angular momentum>[spin-one-half] <massless particle>[massless] fields in the two inequivalent chiral <Spinor representation of the Lorentz group>[spinor representations of the Lorentz group]. Their rotation generators agree, but their <opposite Weyl boost generators>[boost generators] have opposite signs; spatial parity interchanges the two. A positive-energy <particle> in the $\xi$ field has <helicity> $+1/2$, and one in the $\eta$ field has <helicity> $-1/2$. Their antiparticles have the opposite <helicities>, so a <Weyl field> is not a field containing only one <helicity> for both <particles> and antiparticles.
<Neutrinos> provide the observed ultrarelativistic approximation to a left-chiral <particle> and right-helicity antiparticle. However, <neutrino oscillation> establishes nonzero <neutrino mass> differences, so physical <neutrinos> are not all exactly <massless particle>[massless] Weyl <particles>. Chiral <Weyl fields> occur in the <Standard Model>, but an isolated exactly <massless particle>[massless] elementary Weyl <particle> is not established by <neutrino> observations. Whether <neutrino mass> eigenstates are Dirac or Majorana remains a separate question.<sources/neutrino masses, mixing, and oscillations by Particle Data Group (2024)>[Sections 14.2, 14.4 and 14.9.2: massive-neutrino extensions, mass-induced oscillations, and the Dirac–Majorana question]{ref}
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