I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, if we let $S_n$ act on $(\mathbb{C}^N)^{\otimes n}$ by permuting tensor components, so that e.g. $P_{(1 2)} (a \otimes b \otimes c) = b \otimes a \otimes c$, then working in $({\rm End} \mathbb{C}^N)^{\otimes 3}(u, v)$ we have the identity
$\left(u - P_{(1 2)}\right)\left(u + v - P_{(1 3)}\right)\left(v - P_{(2 3)}\right) = \left(v - P_{(2 3)}\right)\left(u + v - P_{(1 3)}\right)\left(u - P_{(1 2)}\right)$
This is used to motivate the definition of an operator $R_{(j k)}(u) = 1 - P_{(j k)} u^{-1}$, the Yang R-matrix, which is then used to express an enormous family of relations on an algebra by multiplying by a matrix of formal power series.
Of course it's straightforward to verify that the above expression holds if we multiply out the terms. That said, it seems considerably less straightforward to me how one would start from $S_3$ and end up at the equation above. Is this just a marvelous ad-hoc construction, or does it belong to some class of examples?