Working in the Dirac-Pauli representation of $\gamma$-matrices, (5.51), show that at high energies
$$
\gamma^5 u^{(s)}=\left(\begin{array}{cc}
\boldsymbol{\sigma} \cdot \hat{\mathbf{p}} & 0 \\
0 & \boldsymbol{\sigma} \cdot \hat{\mathbf{p}}
\end{array}\right) u^{(s)},
$$
where $u^{(s)}$ is the electron spinor of (5.27). That is, show that in the extreme relativistic limit, the chirality operator $\left(\gamma^5\right)$ is equal to the helicity operator; and so, for example, $\frac{1}{2}\left(1-\gamma^5\right) u=u_L$ corresponds to an electron of negative helicity.
Of course, the fact that $\frac{1}{2}\left(1-\gamma^5\right)$ projects out negative helicity fermions at high energies does not depend on the choice of representation. We need only choose a representation if we wish to show explicit spinors. The particular advantage of the Dirac-Pauli representation is that it diagonalizes the energy in the nonrelativistic limit $\left(\gamma^0\right.$ is diagonal), whereas the Weyl representation diagonalizes the helicity in the extreme relativistic limit ( $\gamma^5$ is diagonal).