<s>[INST] 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?
 [/INST] Suppose that there is a set of non-zero integers $(a,b,c,d)$.
First, since $b\not=0$, we have
$$ac=bd\Rightarrow d=\frac{ac}{b}\tag1$$
Moreover, we have
$$ad+bc=ac-bd\Rightarrow (a+b)d=c(a-b).$$
If $a+b=0$, then we have $a-b=0$ because $c\not=0$. This leads $a=b=0$. This is a contradiction. Hence, we have $a+b\not =0$.
Hence, we have $$d=\frac{a-b}{a+b}c\tag2$$
Hence, since $c\not=0$, from $(1),(2)$ we have
$$\frac{ac}{b}=\frac{a-b}{a+b}c\Rightarrow ac(a+b)=bc(a-b)\Rightarrow a^2+b^2=0\Rightarrow a=b=0.$$
This is a contradiction. Hence, we know that there is no sets of non-zero integers $(a,b,c,d)$.
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