spin AM expectation values $$\begin{aligned}
 \langle L_y \rangle &\approx& \frac{-3 \hbar \xi}{\Delta_{lxz/yz,z^2}} \\
 \langle S_y \rangle &\approx& -\frac{3}{4} \hbar \left( \frac{\xi}{\Delta_{lxz/yz,z^2}} \right)^2.
 \label{eq:ham_R_lxz/yz_correction_lysy}
 \end{aligned}$$ for an eigenstate in $x$ direction can be obtained only from the eigenstates with the first-order correction in $\xi$ which leads to hybridization with the $d_{z^2}$ manifold. Thus, the orbital dominant AM texture in the lower $d_{xz/yz}$ bands comes from the inter-band coupling to the $d_{z^2}$. Similarly, we find that the orbital dominant AM texture in the upper $d_{xz/yz}$ bands originates from the inter-band coupling to the $d_{xy}$ and the $d_{x^2-y^2}$ manifolds.

Near the $\mathrm{X}=(\pi,0)$ point, the effective Hamiltonian is $$\begin{aligned}
 \mathcal{H}_{\mathrm{eff}}
 &=&h_{yz}(\vec{k})I_{2\times 2}
 +\left[\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}}+\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}\right]
 \sigma_x k_y -\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}\sigma_y k_x,
\label{eq:ham_eff_X_yz}\end{aligned}$$ where $\tilde{\Delta}_{yz,z^2}=2(t_{\pi}-t_{\delta})-\delta_2$, $\tilde{\Delta}_{yz,x^2-y^2}=2(t_{\pi}-t_{\delta})-\delta_3$, $\tilde{\Delta}_{yz,xy}=2(t_{\pi}-t_{\delta})-\delta_1$, and the Pauli matrices describe the subspace defined by $\{ | d_{yz}\uparrow\rangle, | d_{yz}\downarrow\rangle \}$, and $(k_x, k_y)$ is a local coordinate with respect to $(\pi,0)$. Here, the splitting terms are mixture of Rashba and linear Dresselhaus terms, which are of the form $$\begin{aligned}
\mathcal{H}_{\mathrm{splitting}}=A\sigma_x k_y -B\sigma_y k_x,\end{aligned}$$ with $A=\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}}+\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}$ and $B=\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}$. If we rotate the local coordinate by $\pi/4$ about $k_z$ axis, the splitting terms become $$\begin{aligned}
\mathcal{H}_{\mathrm{splitting}}&=&\frac{A+B}{2}(\sigma_x k_y -\sigma_y k_x)
+\frac{A-B}{2}(\sigma_x k_x -\sigma_y k_y) 