Have I found a counterexample to Noether-Skolem? (No, but I am confused...)
Solution 1:
As indicated in the comments something is wrong with the isomorphism from $(1,-4)_K$ to $(1,1)_K$.
Would $\phi:i_1\mapsto i_2$, $j_1\mapsto j_2(1+\sqrt5 i_2)$ work? Then you have $$ \phi(j_1i_1)=j_2(1+\sqrt 5 i_2)i_2=j_2i_2(1+\sqrt5 i_2)=-i_2j_2(1+\sqrt5 i_2)=\phi(-i_1j_1) $$ as well as $$ \phi(-4)=\phi(j_1^2)=j_2(1+\sqrt5 i_2)j_2(1+\sqrt5 i_2)=j_2^2(1-\sqrt5i_2)(1+\sqrt5i_2)=j_2^2(1-5i_2^2)=-4 $$ as you should.