<s>[INST] Find non-zero real numbers $a,b,c,d$ such that $a^2+c^2=b^2+d^2$ and $ab+cd=0$. There are the obvious cases where $a=c=0$ and $b=d$. Are there any cases for this where all four numbers are non-zero? 
Note, this problem is equivalent to saying the matrix $A^{-1}JA$ is a rotation invariant linear complex structure where $A=\begin{pmatrix}a & b\\c & d\end{pmatrix}\in GL(2,\mathbb{R})$ and $J=\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$ is the usual linear complex structure. 
 [/INST] $$d=-\frac{ab}{c}$$ and we obtain:
$$a^2+c^2=b^2+\frac{a^2b^2}{c^2}$$ or
$$(a^2+c^2)(b^2-c^2)=0.$$
If $b=c$ then we get $d=-a$.
if $b=-c$ then we get $d=a$.
Id est, we got the answer:
$$\{(a,b,b,-a),(a,b,-b,a)|a\neq0,b\neq0\}$$
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