I have come across a lot of decompositions of tensors when I am reading mean curvature flow by Huisken https://projecteuclid.org/download/pdf_1/euclid.jdg/1214438998. For example, on page 4 in order to get an inequality between the gradient of the second fundamental form and the gradient of the mean curvature, he decomposed as $$\nabla_i h_{jk}=E_{ijk}+F_{ijk}$$ where $E_{ijk}=\frac{1}{n+1}(\nabla_i H g_{jk}+\nabla_j H g_{ik}+\nabla_k H g_{ij})$.
Also on the next page, he decomposed an equation as $$|\nabla_ih_{kl}H-\nabla_iH h_{kl}|^2=|\left(\nabla_ih_{kl}H-\frac{1}{2}(\nabla_i H h_{kl}+\nabla_k H h_{il})\right)-\frac{1}{2}(\nabla_i H h_{kl}-\nabla_k H h_{il})|$$ It seems all these decompositions give two orthogonal parts. I am just curious about how to achieve this decomposition in general. I think it's not the symmetric and antisymmetric decomposition of tensors since the first one is already symmetric by the Codazzi equation. I have seen similar things in different context. It's probably something fundamental that I don't know. Btw, what is $F_{ijk}$? Thanks for your help.