Considering $z=a+bi+cj$ ($a,b,c\in\mathbb{Z}$) and $w=d+ei+fj$ ($d,e,f\in\mathbb{Z}$) and the property of complex numbers that $|zw|=|z||w|$. If the rule of multiplication $zw$ is defined such that integer coefficients in $z$ and $w$ will produce integer coefficients in $zw$, then
$$ zw=\alpha+\beta i+\gamma j$$ where $\alpha,\beta,\gamma\in\mathbb{Z}$. Using the property $|zw|=|z||w|$, we can show that $$ (a^2+b^2+c^2)(d^2+e^2+f^2)=\alpha^2+\beta^2+\gamma^2 $$ But this is not true. How can I prove that this last identity does not exist for integers.
If there were such an identity, then if $x$ and $y$ is a sum of $3$ squares, $xy$ would also be a sum of $3$ squares. However, $3$ and $5$ are each the sum of $3$ squares, but $15$ is not.