\operatorname{Tr}[ \mathcal{V}_{{\boldsymbol{\mathbf{k}}}}^\dagger \sigma_0 \otimes \alpha_i \mathcal{V}_{{\boldsymbol{\mathbf{k}}}} \sigma_k \otimes \alpha_j \mathcal{V}_{{\boldsymbol{\mathbf{k}}}}^\dagger \sigma_0 \otimes \alpha_i \mathcal{V}_{{\boldsymbol{\mathbf{k}}}} \sigma_k \otimes \alpha_j