for $\alpha^\mu$ and $\gamma^\mu$ in BC and PL respectively combining the contributions from nearest-neighbor, second-neighbor, and third-neighbor terms (i.e., $\alpha=\sum_n \alpha_n$, $\gamma = \sum_n \gamma_n$): $$\begin{aligned}
&& \alpha^{BC}_{y,a~a=1\sim16} = \frac{9}{2}J_3 - 2 K_3, \\
&& \alpha^{BC}_{y,1~4} = \alpha^{BC}_{y,9~12}= \frac{K_1}{2} e^{-i k_y},\\
\nonumber && \alpha^{BC}_{y,1~5} = \alpha^{BC}_{y,2~3}= \alpha^{BC}_{y,3~7}= \alpha^{BC}_{y,5~6}= \alpha^{BC}_{y,6~10}=\alpha^{BC}_{y,7~8}=  \alpha^{BC}_{y,8~12}=\alpha^{BC}_{y,9~13}=  \alpha^{BC}_{y,10~11}= \alpha^{BC}_{y,11~15}= \alpha^{BC}_{y,13~14}=\alpha^{BC}_{y,15~16}=  \frac{K_1}{2},\\ \\
&& \alpha^{BC}_{y,1~6} = \alpha^{BC}_{y,2~7}=\alpha^{BC}_{y,3~8}= \alpha^{BC}_{y,5~10}= \alpha^{BC}_{y,6~11}=  \alpha^{BC}_{y,7~12}= \alpha^{BC}_{y,9~14}= \alpha^{BC}_{y,10~15}= \alpha^{BC}_{y,11~16}=  \frac{K_2}{2},\\
&& \alpha^{BC}_{y,1~16} = \frac{1}{2}K_2 e^{-i (k_x + k_y)}, \\
&& \alpha^{BC}_{y,2~14}= \alpha^{BC}_{y,4~16}= \frac{K_1}{2} e^{-i k_x},\\
&& \alpha^{BC}_{y,2~13}= \alpha^{BC}_{y,3~14}= \alpha^{BC}_{y,4~15}= \frac{K_2}{2} e^{-i k_x},\\
&& \alpha^{BC}_{y,4~5}= \alpha^{BC}_{y,8~9}= \alpha^{BC}_{y,12~13}= \frac{K_2}{2} e^{i k_y},  \end{aligned}$$ where $\alpha^{BC}_{y, ab} = (\alpha^{BC}_{y, ba})^*$ and rest of the elements are zero. Similarly, $$\begin{aligned}
&& \gamma^{BC}_{y,1~2} =  \gamma^{BC}_{y,2~6}=  \gamma^{BC}_{y,3~4} = \gamma^{BC}_{y,4~8} =  \gamma^{BC}_{y,5~9}=  \gamma^{BC}_{y, 6~7}=  \gamma^{BC}_{y,7~11} =  \gamma^{BC}_{y, 9~10} =  \gamma^{BC}_{y,10~14}=  \gamma^{BC}_{y, 11~12} =  \gamma^{BC}_{y, 12~16}=  \gamma^{BC}_{y,14~15}=\frac{K_1}{2},~~~\\
&&  \gamma^{BC}_{y, 1~3} = \gamma^{BC}_{y,2~4}=\gamma^{BC}_{y,5~7} = \gamma^{BC}_{y, 6~8} = \gamma^{BC}_{y,9~11} = \gamma^{BC}_{y,10~12}=\gamma^{BC}_{y,13~15} = \gamma^{BC}_{y,14~16} = \frac{K_3}{2} \left( 1 + e^{-i k_y}\right),\\
&& \gamma^{BC}_{y,1~8} = \gamma^{BC}_{y,5~12}= \gamma^{BC}_{y,9~16}=\frac{K_2}{2} e^{-i k_y},\\
&& \gamma^{BC}_{y,1~9} = \gamma^{BC}_{y,2~10} = \gamma^{BC}_{y,3~11} = \gamma^{BC}_{y,4~12} = \gamma^{BC}_{y,5~13} = \gamma^{BC}_{y,6~14} = \gamma^{BC}_{y,7~15} = \gamma^{BC}_{y,8~16} = \frac{K_3}{2}\left(1 + e^{-i k_x} \right),\\
&& \gamma^{BC}_{y,1~13} = \gamma^{BC}_{y,3~15} = \frac{K_1}{2} e^{-i k_x},\\
&& \gamma^{BC}_{y,1~14} = \gamma^{BC}_{y,2~15} = \gamma^{BC}_{y,3~16} = \frac{K_2}{2} 