Proof of Euler's four-square identity by expansion

Proof

We expand both sides, using the laws of a commutative ring.

1. Call the four expressions on the right $z_1, z_2, z_3, z_4$, in the order written. Each $z_k$ is a sum of four terms $\pm a_i b_j$.
2. The index pairs $(i, j)$ in the terms are: in $z_1$, $(1,1), (2,2), (3,3), (4,4)$; in $z_2$, $(1,2), (2,1), (3,4), (4,3)$; in $z_3$, $(1,3), (2,4), (3,1), (4,2)$; in $z_4$, $(1,4), (2,3), (3,2), (4,1)$.
3. So each of the $16$ pairs $(i, j)$ occurs in exactly one $z_k$, and in exactly one term.
4. For $t_1, \dots, t_4$ in a commutative ring, $(t_1 + t_2 + t_3 + t_4)^2 = \sum_i t_i^2 + 2\sum_{i < l} t_i t_l$.
5. By steps 3 and 4, the squares of the terms add up to $\sum_{i, j} a_i^2 b_j^2$. This is the left side $(\sum_i a_i^2)(\sum_j b_j^2)$.
6. It remains to show that the cross terms $2 t_i t_l$ cancel. Each cross term is $\pm 2\, a_i a_l b_j b_r$ with $i \ne l$ and $j \ne r$. We list the coefficients of each such monomial, without the factor $2$.
7. $a_1a_2b_1b_2$: $-1$ from $z_1$ and $+1$ from $z_2$. $a_1a_2b_3b_4$: $-1$ from $z_3$ and $+1$ from $z_4$.
8. $a_1a_3b_1b_3$: $-1$ from $z_1$ and $+1$ from $z_3$. $a_1a_3b_2b_4$: $+1$ from $z_2$ and $-1$ from $z_4$.
9. $a_1a_4b_1b_4$: $-1$ from $z_1$ and $+1$ from $z_4$. $a_1a_4b_2b_3$: $-1$ from $z_2$ and $+1$ from $z_3$.
10. $a_2a_3b_2b_3$: $+1$ from $z_1$ and $-1$ from $z_4$. $a_2a_3b_1b_4$: $+1$ from $z_2$ and $-1$ from $z_3$.
11. $a_2a_4b_2b_4$: $+1$ from $z_1$ and $-1$ from $z_3$. $a_2a_4b_1b_3$: $-1$ from $z_2$ and $+1$ from $z_4$.
12. $a_3a_4b_3b_4$: $+1$ from $z_1$ and $-1$ from $z_2$. $a_3a_4b_1b_2$: $+1$ from $z_3$ and $-1$ from $z_4$.
13. Each $z_k$ has $6$ cross terms, so there are $24$ in all. Steps 7 to 12 account for all $24$, in $12$ pairs with opposite signs.
14. For example, in $z_1$ the terms $a_1b_1$ and $-a_2b_2$ give $-2a_1a_2b_1b_2$. In $z_2$ the terms $a_1b_2$ and $a_2b_1$ give $+2a_1a_2b_1b_2$. The other lines are read off in the same way.
15. So all cross terms cancel, and $z_1^2 + z_2^2 + z_3^2 + z_4^2$ equals the left side.