Resistive evolution of the magnetic second-moment tensor ($D_{t}M-LM-M{L}^T={\eta }_m{\nabla }^{2}M-2{\eta }_m{\partial }_n\mathbf{B}{\partial }_n{\mathbf{B}}^T$)

For constant magnetic diffusivity, apply the product rule to $M_{ij}=B_i{B}_j$ and the incompressible resistive induction equation. The identity ${\nabla }^2(B_i{B}_j)=B_i{\nabla }^2{B}_j+B_j{\nabla }^2{B}_i+2{\partial }_n{B}_i{\partial }_n{B}_j$ proves the displayed formula. Omitting the gradient-product term gives the wrong tensor evolution. Its average generally requires information beyond the averaged magnetic second-moment tensor.