I asked this question in "math.stackexchange" but I did not get any response, so I put it here, maybe someone can help.

Is it possible to write identity for $$ \{x(y^2-z^2)-y\}.\{u(v^2-w^2)-v)\}=a(b^2-c^2)-b $$ in integers similar to the identity $$ (x^2+y^2)(u^2+v^2)=a^2+b^2,\qquad a=xu+yv,\qquad b=xu-yv. $$

If possible, what can we choose for $a,b,c$ to be in terms of $x,y,z,u,v,w$?