I was toying around with central simple algebras over a field $K$ today and thought that I should try to verify Noether-Skolem's theorem that any automorphism of such must be inner. So, let us take $K = \mathbb{Q}[\sqrt{5}]$, and the quaternion algebra $(1,-4)_K$, i.e. $i_1^2=1$,$j_1^2=-4$ and $i_1j_1=-j_1i_1$. Now, we can see that this is isomorphic to $M_2(K)$, the matrix ring, where the isomorphism is given by $$j_1 \rightarrow \pmatrix{0 & -4 \\ 1 & 0},$$ $$i_1 \rightarrow \pmatrix{1 & 0 \\ 0 & -1 }.$$ Call $\psi$ the isomorphism $(1,-4)_K \rightarrow M_2(K)$. Now, we can see that as a quaternion algebra, $(1,-4)_K \cong (1,1)$ ( call the generators here for $i_2$ , $j_2$) by the isomorphism $$i_1 \rightarrow i_2,$$ $$j_1 \rightarrow j_2(1+i_2)$$ and we call this isomorphism by $\phi$. Now, $(1,1)_K$ is isomorphic to $M_2(K)$ as well, let $\sigma:(1,1)_K \rightarrow M_2(K)$ be the isomorphism (which is of the same "form" as $\psi$). Now, this induces an automorphism: $\sigma \circ \phi \circ \psi^{-1}: M_2(K) \rightarrow M_2(K)$, which takes $$e= \pmatrix{1 & 0 \\ 0 & -1 }$$ onto itself and takes $$f= \pmatrix{0 & -4 \\ 1 & 0}$$ onto $\sigma(j(1+i))$. Now, I can not get this to be an inner automorphism! I get that all entries must be zero, and this is clearly nonsense. So please, mathstackexchange, before I lose my mind (that's an exaggeration, but it is annoying), how can this be resolved?
Update (A concrete description of $\sigma(j(i+1))$
In the comments I got asked whether I could write out what $\sigma(j(i+1))$ was. I left it out on purpose, since I believe this might be where the error lies. However, this is how my thinking goes: $$\sigma(j(i+1))= \sigma(j)\sigma(i+1) = \pmatrix{0 & 1 \\ 1 & 0 } [ \pmatrix{1 & 0 \\ 0 & -1 } + \pmatrix{1 & 0 \\ 0 & 1}].$$
Multiplying this, I get $$\pmatrix{ 0 & 0 \\ 2 & 0 }.$$
So if we assume that we have an inner automorphism: we should have $$\pmatrix{a & 0 \\ 0 & b } \pmatrix{0 & -4 \\ 1 & 0 } = \pmatrix{0 & 0 \\ 2 & 0} \pmatrix{a & 0 \\ 0 & b}$$
but this can not hold! So, some of my assumptions should be wrong, or some step, but I really can't see which one. I suspect that it lies in the step where I multiply by a norm. But it should hold, but maybe my isomorphism isn't an isomorphism. But there should be one, and I am quite sure that that one is correct!
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.