= Solution
The two <Weyl spinors> transform in the inequivalent left- and <Chirality>[right-chiral] representations of the proper <Lorentz group>. They are selected by $P_{L,R}=(1\mp\gamma^5)/2$ and have different <Lorentz boost>[boost] transformation laws; <parity> exchanges the two. A massive <Dirac equation> couples them, whereas in the massless limit they can propagate independently.
For the particular $u$ printed in this paper, transforming its supplied Dirac $\gamma^5$ gives
$$
\boxed{\gamma_W^5=u\begin{pmatrix}0&I\\I&0\end{pmatrix}u^{-1}=\begin{pmatrix}I&0\\0&-I\end{pmatrix}.}
$$
Thus the upper block is <Chirality>[right chiral] and the lower block <Chirality>[left chiral] in this convention. Many references use the opposite block order; neither convention changes physics, but they must not be mixed. <Chirality> is not generally the same as <helicity> for a massive <electron>. The weak <charged current> selects the <Chirality>[left-chiral] <electron> component, reducing to negative <helicity> at high <energy>.
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