I am trying to understand following lemma from Class Field Theory by Milne- https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.3.2.2
I want to prove that if $L/K $ is galois of finite degree $n$ then $H^2(L/K) $ contains subgroup canonically isomorphic to $(1/n) \mathbb Z /\mathbb Z $.
They consider the above diagram. Here $H^2(L/K) $ means $H^2(Gal(L/K),L^ {\times}) $.
The inflation map is claimed to be injective. This follows as:
By Hilbert's satz 90, $H^1(K^{al}/K)=0 $. So we can apply the inflation-restriction exact sequence to get that the Inf maps are injective. I actually don't quite understand why the rightmost square, which is also second square from left, commutes? I understand it must just be unravelling the definitions but I am not very confident. Is it ultimately because restriction of field automorphism from $L$ to $E$ and then $E$ to $K$ is same as $L$ to directly $K$ for tower $K \subset E \subset L $.
This square induces the leftmost vertical map and it is injective as the inflation maps are.
Feel free to give any reference and any help is appreciated.