Lemma for the construction of the reciprocity map

Question

I do not understand the highlighted part in the following proof, namley that $N(\tilde x)=1$.
To give some context, this proof is taken from Neukirch's Algebraic Number Theory, where $\tilde K$ indicates the maximal unramified extension of $K$ (and the same for $L$).

It would suffice to know that $\sigma_n$ and $\tau_i$ commute, but I don't know if it's true in general..

Answer 1

Ok, solved, they commute because $\sigma\in G(\tilde{L}\mid L)$ and $\tau_i\in G(\tilde L\mid\tilde K)$, which are two normal subgroups of $G(\tilde L\mid K)$ whose intersection is trivial.