This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet’s Lemma for \({{\,\textrm{GL}\,}}_2\) in the residually indistinguishable case. We suppose we are given a Galois representation taking values in the total ring of fractions of a complete reduced Noetherian local ring \(\textbf{T}\) , such that the characteristic polynomial of the representation is reducible modulo some ideal \(I \subset \textbf{T}\) . We assume that the two characters that arise are congruent modulo the maximal ideal of \(\textbf{T}\) . We construct an associated Galois cohomology class valued in a \(\textbf{T}\) -module that is “large” in the sense that its Fitting ideal is contained in I. We make some simplifying assumptions that streamline the exposition—we assume the two characters are actually trivial, and we ignore the local conditions needed in arithmetic applications.