How to justify Einstein notation manipulations without explicitly writing sums?

Question

In calculating the expression for the coordinates of the Lie Bracket of two vector fields, one has to "interchange the roles of the dummy indices $i$ and $j$ in the second term" (p.187, Lee Introduction to Smooth Manifolds) i.e. justify the following equality: $$X^j \frac{\partial Y^i}{\partial x\ ^j} \frac{\partial f}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i}\frac{\partial f}{\partial x\ ^j} \overset{?}{=} X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} - Y^j\frac{\partial X^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i}. $$ Now writing out the sums explicitly this is fairly easy to do: $$\sum_i\sum_j \left[X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} - Y^i \frac{\partial X^j}{\partial x^i}\frac{\partial f}{\partial x^j} \right] = \sum_i\sum_j\left[X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} \right] - \sum_i\sum_j\left[ Y^i \frac{\partial X^j}{\partial x^i}\frac{\partial f}{\partial x^j} \right] \\ = \sum_i\sum_j\left[X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} \right] - \sum_j\sum_i \left[ Y^j \frac{\partial X^i}{\partial x^j}\frac{\partial f}{\partial x^i} \right] = \sum_i\sum_j\left[X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} \right] - \sum_i\sum_j \left[ Y^j \frac{\partial X^i}{\partial x^j}\frac{\partial f}{\partial x^i} \right] \\ = \sum_i\sum_j \left[X^j \frac{\partial Y^i}{\partial x\ ^j}\frac{\partial f}{\partial x^i} - Y^j \frac{\partial X^i}{\partial x^j}\frac{\partial f}{\partial x^i} \right] .$$ However, writing out all of these sums is fairly laborous and defeats the purpose of using Einstein notation in the first place.

Question: Is there a list somewhere of allowed manipulations using Einstein notation? I would like to use such a list to rigorously justify manipulations like the above using Einstein notation in the future with a clean conscience.

I could probably supply the proofs writing out the sums explicitly myself, so the list of allowed manipulations doesn't need to come with proofs for all of the rules.

Note: This is related to a previous question of mine, where I asked (essentially) whether and if so which and how many manipulations using Einstein notation require finiteness of the index sets in order to be justified. Note that the above calculation is another example where the finiteness of the indexing sets is appealed to implicitly in order to justify switching the order of summation in the second-to-last step (the third-to-last step consists simply of renaming variables).

Answer 1

The same rule you used to transform $i \leftrightarrow j$ in the second line of the second equation also applies in Einstein's notation

$$ u^i v_i = u^jv_j $$

which is to say that you can call dummy indices the way you like but making sure that only one upper-index sums with only one lower-index. For instance, you can make this

$$ x^i A_{ij} y^j = x^j A_{ji} y^i $$

but you cannot do this

$$ x^i A_{ij} y^j = x^i A_{ii} y^i $$

Answer 2

It is not entirely clear what you are asking but let me point out that the calculation you wrote down is correct even if you erase all the $\Sigma$s, so you could have written a much shorter calculation to justify this. What is in the background is essentially the distributivity law for the real numbers as well as associativity of multiplication. Of course you can exploit commutativity of multiplication also but this does not seem to be used in the particular calculation you presented.

To convince yourself that $u^i_j v_i^j + w^j_i x^i_j = u^i_j v_i^j + w^i_j x_i^j$ is a legitimate procedure, first change the names of the variables in the second summand: $$ u^i_j v_i^j + w^j_i x^i_j = u^i_j v_i^j + w^p_q x_p^q. $$ Now since both $p$ and $q$ are dummy variables, they can be changed respectively to $i$ and $j$ (in that order), so that we get $$ u^i_j v_i^j + w^j_i x^i_j = u^i_j v_i^j + w^p_q x_p^q=u^i_j v_i^j + w^i_j x_i^j. $$ When one uses the Einstein summation convention, the $\Sigma$ signs carry no additional information at all; in particular it cannot be said that something can be shown with the $\Sigma$s but not without them.