condition for tuple of non-zero integers $(a,b,c,d)$ such that $ad+bc=ac-bd=ab+cd=a^2-b^2+c^2-d^2=0$ As the title says, what would be the condition for tuple of non-zero integers $(a,b,c,d)$ such that $ad+bc=ac-bd=ab+cd=a^2-b^2+c^2-d^2=0$? Would there be infinitely many tuples that satisfy the condition above?

Let $i = \sqrt{-1}$. Then, $(a+bi)(c+di) = (ac-bd)+(ad+bc)i = 0+0i = 0$, so either $a+bi = 0$ or $c+di = 0$, i.e. either $a = b = 0$ or $c = d = 0$. 
If $a = b = 0$, then we are left with $cd = c^2-d^2 = 0$, from which we get $c = d = 0$. 
If $c = d = 0$, then we are left with $ab = a^2-b^2 = 0$, from which we get $a = b = 0$. 
Therefore, the only solution to $ad+bc=ac-bd=ab+cd=a^2-b^2+c^2-d^2=0$ is the trivial solution $(a,b,c,d) = (0,0,0,0)$. 
