Bosonic quadratic Nambu normalization ($H=C-\frac{1}{2}{\sum }_k{A}_k+\frac{1}{2}{\sum }_k{\Psi }_k^{†}\left(\begin{array}{cc}{A}_k& B_k\\ B_k& A_k\end{array}\right){\Psi }_k$)

For real even coefficients, ${\Psi }_k=(a_k,a_{-k}^{†}{)}^{\mathsf{T}}$ and a full-zone sum, this expression equals $C+{\sum }_k[A_k{a}_k^{†}{a}_k+B_k(a_k{a}_{-k}+a_k^{†}{a}_{-k}^{†})/2]$. The constant subtraction follows from $a_{-k}{a}_{-k}^{†}=1+a_{-k}^{†}{a}_{-k}$. Pairing redundancy and normal ordering determine both factors together. Switching to half a zone requires grouping both members of each pair; changing only a diagonal factor is inconsistent.