My question is based on the answer of @Travis Willse in the discussion
Can we express sum of products as product of sums?
Given the expression,
$$ \left[\sum\limits_{i=1}^n (a_i+b_i)e_i\right] \left[\sum\limits_{j=1}^n (c_j+d_j)e_j\right]$$, can I rewrite this to $$ \sum\limits_{i=1}^n \sum\limits_{j=1}^n (a_i+b_i)(c_j+d_j)e_i e_j = \sum\limits_{i=1}^n \sum\limits_{j=1}^n (a_i c_j+a_i d_j + b_i c_j + b_i d_j) e_i e_j $$
Note that $e_i$ and $e_j$ are the same numbers in both sums. But since I multiply two sums, I have to give them a different index in both sums, right?