Commutativity of Contractions I'm working through Pavel Grinfeld's book 'Introduction to Tensor Analysis and the Calculus of Moving Surfaces' and I am having trouble understanding the example given in section 4.9.2 - Commutativity of Contractions:

For an illustration, consider an object $T_{kl}^{ij}$ with four indicies and the double contraction $T_{ij}^{ij}$. The expression $T_{ij}^{ij}$ can be interpreted in three ways. First, we can imagine that contraction on i takes place first, yielding the object $U_l^j = T_{il}^{ij}$. Fully expanding, $U_l^j$ is the sum
$U_l^j = T_{1l}^{1j} + T_{2l}^{2j} + T_{3l}^{3j}$

What I don't understand is the l that now appears in $U_l^j$ instead of j because surely now there can be no contraction on j because there are different indices on the top and bottom? Why is there the need to introduce the index l?
 A: I think you are somehow confused by the fact that $j$ and $l$ are seemingly different: they could be but what if they're equal then you can contract them.
We are given a tensor of type $(2,2)$, $T^{ij}_{kl}$, element of some vector space $V \otimes V\otimes V^*\otimes V^*$, with 2 contravariant elements and 2 covariant elements.
The point of this short paragraph in the book is to illustrate that we can first contract $i$ and $k$ (that is setting $i=k$), getting a mixed tensor of type $(1,1)$ and then contract $j$ and $l$ (that is setting $j=l$) or first contract $j$ and $l$ (again setting $j=l$) and then after that $i$ and $k$ (setting $i=k$), that is: contractions commute.
Suppose first that $i=k$ and contract these indices. One then obtain a $(1,1)$ mixed tensor with one covariant element and one contravariant element and it can be written as $$U^j_l= T^{1j}_{1l}+T^{2j}_{2l}+T^{3j}_{3l}.$$
Now next to operate the second contraction, let $j=l$, and smilarly one gets the following:
$$T^{11}_{11}+T^{12}_{12}+T^{13}_{13}+T^{21}_{21}+T^{22}_{22}+T^{23}_{23}+T^{31}_{31}+T^{32}_{32}+T^{33}_{33}.$$
You can do the contractions either way first. Alternatively you can do the contractions simultneously: just write down all possible combinations for $T^{ij}_{ij}$ after having set $i=k$ and $j=l$.
