2^nd + 3^rd

1^st + 4^th

$\begin{array}{l}{\partial }_i{\partial }^j{\nabla }_j{\xi }^i-{\partial }_i{\partial }^i{\nabla }_j{\xi }^j\\ ={\partial }_i{\nabla }_j\left({\partial }^j{\xi }^i-{\partial }^i{\xi }^j\right)\end{array}$

${\partial }_i{\partial }^j{\nabla }^i{\xi }_j-{\partial }_i{\partial }^i{\nabla }^j{\xi }_j$

Exchange of i and j in the first term

$={\partial }_j{\partial }^i{\nabla }^j{\xi }_i-{\partial }_i{\partial }^i{\nabla }^j{\xi }_j={\partial }^i{\nabla }^j\left({\partial }_j{\xi }_i-{\partial }_i{\xi }_j\right)$

or

$={\partial }_i{\nabla }_j\left({\partial }^j{\xi }^i-{\partial }^i{\xi }^j\right)$