I am confused by the indicial term $u_{j,ij}$ and cannot find it treated in discussions of tensor/indicial/Einstein notation even though it is an important term in linear elasticity. Working off context, it appears to be the gradient of the divergence but I do not see how. I can see that $u_{j,j}$ would be the divergence, but I don't understand how putting an $i$ in the middle takes the gradient of the divergence, nor do I understand how to sum the $j$ indices with the $i$ in the middle of them. Can anyone expand the summation process for this term and show its equivalent in vector notation?
I also wish I had a more intuitive understanding of why grad(div u)) would play such a prominent role in elastic equilibrium but that is probably more of a physics question.
Comment turned into answer per request.
$$\begin{align}u_{j,ij} &= \sum_{j=1}^3 \frac{\partial}{\partial x_j}u_{j,i} = \sum_{j=1}^3 \frac{\partial^2}{\partial x_j\partial x_i} u_j = \sum_{j=1}^3 \frac{\partial^2}{\partial x_i\partial x_j} u_j = \sum_{j=1}^3 \frac{\partial}{\partial x_i} u_{j,j}\\ &= \frac{\partial}{\partial x_i} \nabla\cdot u = \hat{e}_i \cdot \nabla (\nabla\cdot u) \end{align}$$
The key is partial derivatives $\frac{\partial}{\partial x_i}$ commute among themselves.