Originally Posted by

**simplependulum** If $\displaystyle X $ and $\displaystyle Y $ can be written as the sum of 8 squares , then their product is also the sum of 8 squares .

that means

$\displaystyle ( a^2 + b^2 + c^2 + d^2 + e^2 + f^2 +g^2 + h^2)( A^2 + B^2 + C^2 + D^2 + E^2 + F^2 + G^2 +H^2) =$

$\displaystyle \alpha^2 + \beta^2 + \gamma^2 + \delta^2 + \epsilon^2 + \zeta^2 + \eta^2 + \theta^2 $

Show that the relationship between $\displaystyle a,b,c,d...A,B,C,D...\alpha , \beta , \gamma , \delta ... $ is :

$\displaystyle \left(\begin{array}{cccccccc}a&b&c&d&e&f&g&-h\\-b&a&-d&c&-f&e&h&g\\c&-d&-a&b&g&h&-e&f\\d&c&-b&-a&h&-g&f&e\\-e&f&g&h&a&-b&-c&d\\-f&-e&h&-g&b&a&d&c\\g&h&e&-f&-c&d&-a&b\\h&-g&f&e&-d&-c&b&a\end{array}\right)$$\displaystyle \left( \begin{array}{c}A\\B\\C\\D\\E\\F\\G\\H\end{array}\ right) = \left(\begin{array}{c}\alpha\\ \beta\\ \gamma \\ \delta \\ \epsilon \\ \zeta \\ \eta \\ \theta\end{array}\right)$