I already asked this question on MSE here https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra, but still received no answer. I hope I will be more lucky here.
When you teach algebra to students, it's often easy to find cute/direct applications of "big" theorems to motivate how useful these results can be.
For example, group actions, Sylow theorems, or the first isomorphism theorem have nice applications and can provide non trivial results in few lines.
However, I'm struggling to find such applications concerning the Snake Lemma. As the title suggests, I would like applications outside homological algebra, so an answer like "we can use it to prove the $n$-lemma" (pick for $n$ your favorite integer), is not the kind of answer I'm looking for.
Precisely, I would like to know applications of the Snake Lemma, even direct and/or not very sophisticated, but which would have been difficult or lengthy to prove without it.
I found such an example here, which is rather sophisticated: https://math.stackexchange.com/questions/682777/is-an-abelian-group-characterized-by-its-localizations/712351#712351
Maybe you will have other examples , even shorter or simpler?
Thanks for your help !
Greg