 Bosonic quadratic Nambu normalization

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.