I'm currently trying to show that the ring $A =\begin{pmatrix} a & b \\ - \bar{b} & \bar{a} \end{pmatrix} $ is isomorphic to the real quaternions $\mathbb{H}$ as $\mathbb{R}$-algebras.
I've already shown that $A$ is a subring of $M_2(\mathbb{C})$ and that its centre is $$Z(A) = \left\{\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} | a \in \mathbb{R} \right\}. $$
Since we are considering $A$ and $\mathbb{H}$ as $\mathbb{R}$-algebras, we know that there are structure maps $f: \mathbb{R} \rightarrow A$ and $g: \mathbb{R} \rightarrow \mathbb{H}$ s.t. $f(\mathbb{R}) \subset Z(A)$ and $g(\mathbb{R}) \subset Z(\mathbb{H}) = \mathbb{R}$.
From what I can gather, I need to show that there's some function $h: A \rightarrow \mathbb{H}$ which is a bijective homomorphism of $\mathbb{R}$-algebras. And to show it is a homomorphism, I need to show $h \circ f = g$.
And this is where I'm stuck. I haven't been shown an explicit example of how to show two R-algebras are isomorphic, so a bit confused as to how to go about it all. Any help in the right direction would be greatly appreciated! Thanks!
We have: $$ z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{i} \quad \mapsto \quad \mathbf{Z}= a\mathbf{U}+b\mathbf{I}+c\mathbf{J}+d\mathbf{K} \quad a,b,c,d \in \mathbb{R} $$ with: $$ \mathbf{U}= \left( \begin{array}{ccccc} 1&0 \\ 0 &1 \end{array} \right) \qquad \mathbf{I}= \left( \begin{array}{ccccc} i&0 \\ 0 &-i \end{array} \right) \qquad \mathbf{J}= \left( \begin{array}{ccccc} 0&1 \\ -1 &0 \end{array} \right) \qquad \mathbf{K}= \left( \begin{array}{ccccc} 0&i \\ i &0 \end{array} \right) $$ and: $$ \mathbf{I}^2=\mathbf{J}^2=\mathbf{K}^2=\mathbf{I}\mathbf{J}\mathbf{K}=-\mathbf{U} $$ $$ \mbox{det}(\mathbf{Z})= \left | \left( \begin{array}{ccccc} a+ib&c+id \\ -c+id &a-ib \end{array} \right) \right |= a^2+b^2+c^2+d^2=|z|^2 $$