Let $1= \left[ {\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} } \right],\ i= \left[ {\begin{array}{cc} i & 0 \\ 0 & -i \\ \end{array} } \right],\ j= \left[ {\begin{array}{cc} 0 & 1 \\ -1 & 0 \\ \end{array} } \right],\ k= \left[ {\begin{array}{cc} 0 & i \\ i & 0 \\ \end{array} } \right]$. Show that $\alpha= \left[ {\begin{array}{cc} a+bi & c+di \\ -c+di & a-bi \\ \end{array} } \right]$ may be written in the form $$\alpha=a1+bi+cj+dk$$
So is this because $a\cdot 1 + b \cdot i$ is the first spot in $\alpha$? I can see that $i^2, j^2, k^2$ are all going to give me $-1$.
No. It's because\begin{align}\begin{bmatrix}a+bi&c+di\\-c+di&a-bi\end{bmatrix}&=\begin{bmatrix}a&0\\0&a\end{bmatrix}+\begin{bmatrix}bi&0\\0&-bi\end{bmatrix}+\begin{bmatrix}0&c\\-c&0\end{bmatrix}+\begin{bmatrix}0&di\\di&0\end{bmatrix}\\&=a\begin{bmatrix}1&0\\0&1\end{bmatrix}+b\begin{bmatrix}i&0\\0&-i\end{bmatrix}+c\begin{bmatrix}0&1\\-1&0\end{bmatrix}+d\begin{bmatrix}0&i\\i&0\end{bmatrix}\\&=a1+bi+cj+dk.\end{align}