\,\, d_5(k) =
\frac{-1}{2}(2 k_z^2 - k_x^2 -k_y^2) .\end{aligned}$$ It is straightforward to relate the masses used in the main text and the Luttinger $\alpha_i$ parameters. This can be done by expressing the spin operators in terms of gamma matrices, using the equalities $$\begin{aligned}
&& J_x = \frac{\sqrt{3}}{2} \Gamma_{15} - \frac{1}{2} (\Gamma_{23} - \Gamma_{14}) \ ,\nonumber \\
&& J_y = -\frac{\sqrt{3}}{2} \Gamma_{25} + \frac{1}{2} (\Gamma_{13} + \Gamma_{24}) \ ,\nonumber \\
&& J_z = - \Gamma_{34} - \frac{1}{2} \Gamma_{12} \ ,\end{aligned}$$ where $\Gamma_{ab} = \frac{1}{2 i} [\Gamma_a, \Gamma_b]$ is used.

Weyl semimetal
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Here we consider how the Weyl semimetal appears, taking for simplicity the case $\delta =0 , \theta_1 < \theta < \theta_2$. It is straightforward to generalize this to the cases with $\delta\neq 0$. For convenience, we set $m=1/2$. For $\vec{k}=(0,0,k_z)$, the Hamiltonian in the presence of the Zeeman field, $h$ (along the same axis), becomes $$\begin{aligned}
\mathcal{H}(k_z \hat{z})&=& k_z^2 (\frac{5}{4}-J_z^2)-H ( \cos(\theta) J_z +\sin(\theta) J_z^3). \nonumber \end{aligned}$$ Clearly, the energy eigenstates can be labeled by both $k_z$ and $J_z=\pm 1/2, \pm 3/2$. One can readily see that level crossings occur between the two bands with $J_z=1/2$ and $J_z=-3/2$. In the vicinity of these crossing points, the other states 