I am reading about Reidemeister–Schreier rewriting process and I have the following question.
After we find a set of generators for the subgroup we find the set of its relators which are of the form $trt^{-1}$, where $t\in T$ (Schreier transversal) and $r$ are the relators of the group rewritten in terms of the generators of the subgroup.
My question is in the process where we rewrite these relators are we allowed to use that these relations hold in the group and thus in the subgroup?
For example if I have a group $G$ which has a relator $r=[a,b]$ and let $H$ be a subgroup that has as set of generators $S=\{a^{2},b^{2}\}$.
Now in order to write $r$ in terms of $S$ can I use that $a$ and $b$ commute with each other?
No. You may not use identities in the group (as they just might be a consequence of the relator that is rewritten). Formally, the rewriting process takes place in the free group and expresses normal subgroup generators of the kernel generated by relations.
In your example, the rewriting process to $x=a^2,y=b^2$ of rewriting the rule $[a,b]$ should result in a rule that $x$ and $y$ commute, and once you have stored that rule you may use it.
Indeed, when doing rewriting on the computer, what is done is to apply a heuristic of Tietze transformations to the resulting presentation to eliminate redundancies and possibly make use of some dependencies amonst the new generators that were found.