I'm looking for a matrix $M$ with $\det(M)=a^2+b^2+c^2+d^2$ In order to show that $(a^2+b^2+c^2+d^2)(A^2+B^2+C^2+D^2)=  \alpha^2+\beta^2+\gamma^2+\delta^2$ with $a,b,c,d,A,B,C,D,\alpha,\beta,\gamma,\delta \in \mathbb Z$.
I would like to find a matrix with determinant $a^2+b^2+c^2+d^2$ in order to multiply the matrix with another with determinant $A^2+B^+C^2+D^2$.

The matrix, where $i = \sqrt{-1}$,
\begin{pmatrix}
a+ib & -c+id\\
c+id & a-ib
\end{pmatrix}
works. When we multiply two matrices of the above form, we get a matrix of the same form. 
