I have some problems in understanding some details in the proof of Skolem-Noether Theorem from Vignéras' book. The statement is
"Let L, L' be two commutative $k$-algebras inside a quaternion algebra $H/k$. Then each $k$-isomorphism of $L$ onto $L'$ lifts to an inner isomorphism of $H$ and all $k$-automorphisms of $H$ are inner automorphisms."
In the proof Vignéras considers a commutative $k$-algebra $L$ different from $k$ and a non trivial $k$-isomorphism of $L$ in $H$, called $g$. She then defines two actions of $L$ on $H$: $$m.h=mh\quad and \quad m.h=g(m)h$$ where $m\in L$ and $h\in H$. Then she asserts the existence of a linear $k$-endomorphism of $H$, called $z$, s.t. $z(mh)=g(m)z(h)$.
First question: why does this $z$ exist?
Going on in the proof it appears an element $a\not\in L$ and this fact implies $H=L+aL$.
Second question: why $a\not\in L$?
Third question: why $a\not\in L \Rightarrow H=L+aL$?
I couldn't write down all Vignéras' proof but, if you don't have the book, there is a Pdf english version if you search "the arithmetic of the quaternion algebra" on Google. You can also find it at the link:
http://maths.nju.edu.cn/~guoxj/notes/qa.pdf
I noticed a small mistake in this Pdf version: there is written $a\in L$ in place of $a\not\in L$. There might be other mistakes, so be careful.
P.S. I am not english so I hope my writing is acceptable.
P.P.S. I found other versions of Skolem-Noether but I don't think I can apply them cause $L$ is not supposed to be a simple $k$-algebra but just a commutative $k$-algebra.
Thank you in advance :)