It is not very difficult to show that every odd prime number $p$
can be written in exactly $(p+1)/2$ different ways as
$$p=a\cdot b+c\cdot d$$
with $a,b,c,d\in\mathbb N$ satisfying $0\leq c,d\leq \max(c,d)<\min(a,b)$.

Example for $p=23$:
$$1\cdot 23+0\cdot 0,23\cdot 1+0\cdot 0,2\cdot 11+1\cdot 1,11\cdot 2+1\cdot 1, 3\cdot 7+2\cdot 1,3\cdot 7+1\cdot 2,$$
$$7\cdot 3+2\cdot 1,7\cdot 3+1\cdot 2,4\cdot 5+1\cdot 3,4\cdot 5+3\cdot 1,5\cdot 4+1\cdot 3,5\cdot 4+3\cdot 1$$

Since Kleins Viergruppe $\mathbb V$ acts an all such quadruplets $(a,b,c,d)$ 
(by permutating the first two, the last two or the first two and the last two elements), we get an easy proof that every prime 
$p$ congruent to $1$ modulo $4$ must be a sum of squares 
($(p+1)/2$ is then odd and the only fixpoints under the action 
of $\mathbb V$ are of the form $(a,a,c,c)$).

*Does somebody know a reference for this proof?* (It looks a bit like
Zagier's proof which uses also a parity argument for a set acted upon by involutions.)

**Motivation:** This is in fact a variation of
https://mathoverflow.net/questions/405035 .