I has a question reading 'Galois cohomology-Gille-Szamuely'
I has a problem as follows.
Let us define a k-algebra homomorphism $\varphi:(a,b) \to (u^2a,v^2b) $ which assigns $ui$ to $i$ and $vj$ to $j$. $\varphi(ii)=\varphi (a)$ and $\varphi(i)\varphi(i)=uiui=u^2a$ are equal since $\varphi$ is a k-algebra homomorphism so $\varphi (a)$ should be $u^2 a$ which makes $\varphi(i)\varphi(i) =u^2 a =\varphi(ii)$.
$\varphi$ is not identity homomorphism on $k$ because of $\varphi(a)=au^2$.
I think that $\varphi$ should be identity homomorphism on k(as I understand it is a definition of a morphism of quaternion algebras)
my question is "$\varphi$ don't need to be identity homomorphism on k?"