In Babak Majidzadeh Garjani's note Summation Convention and Vector Algebra (link), he seems to imply multiple times that whenever the 'free index analysis' rule is not violated, one can proceed safely. The notes contain no proof of this.
The idea of the analysis is similar to dimensional analysis in physics. The idea is that per example $$ a_{ij}(x_i+x_j) \neq a_{ij}x_i + a_{ij}y_j, $$ due to the fact that setting $z_{ij} = (x_i+x_j)$, we have on the left-hand-side $a_{ij}(x_i+x_j) = a_{ij}z_{ij}$ which is a 'term' (the author's terminology) that has no free indices while on the right-hand-side we have two terms but even the first of them, $a_{ij}x_i$, has $j$ as a free index. By this 'free index analysis', the equality cannot hold.
That is fine, but as I explained in the first paragraph, the author seems to imply that the converse is true: Whenever the free index analysis holds, the manipulation is valid. He uses this later in a manipulation such as $$ a_{ij}x_iy_j + a_{ji}x_iy_j = (a_{ij} + a{ji})\;x_iy_j. $$ This is an identity stated explicitly in Schaum's outline of theory and problems of tensor calculus by David Kay.
I was not able to find anywhere a proof of this identity. I think this is usually given as an exercise. But what would be more useful is a proof of the general principle in bold. Is it true? Is it treated in detail somewhere?