Élie Cartan's son Henri Cartan was also a great mathematician. As well as some great papers, that Cartan did a lot of collaborative work, including Bourbaki and the Cartan seminar. A lot of things are also named after him, for instance the Cartan-Eilenberg resolution.
I can't access the Cartan lemma in Freitag and Kiehl. I see no direct connection between your Cartan lemma in complex analysis, which is a lower bound on the norm of the value of a complex polynomial at a non-root; and your Cartan lemma on decompositions of non-Archimedian associative in normed rings.
Élie Cartan's son Addendum: Thanks to Mohan's answer, I now know that the bound for complex polynomials is due to Henri Cartan was in 1928. It looks like the Cartan lemma on normed rings is also a great mathematiciandue to Henri Cartan in 1940, but his biggest contributions were and was originally applied in complex analysis even though it is a result in ring theory. So maybe my historical review is not of all that well informed for the same type as those of his dadspecific question. That Cartan did more collaborative workAnd, including Bourbaki although I still don't have access to Freitag and the Cartan seminarKiehl, the result there could be a Cartan's Lemma in sheaf cohomology which would lead is also due to fewer things named after himHenri Cartan. But some things are named after himSo maybe my first paragraph playing up Élie Cartan in the first paragraph is not all that well informed for this question, although he is also certainly responsible for instance the Cartan-Eilenberg resolutionvarious "Cartan's Lemmas".