Product of Sums of Four Squares

Theorem
Let $$a, b, c, d, w, x, y, z$$ be numbers.

Then:

$$ $$ $$ $$ $$

Corollary
Let $$a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n, c_1, c_2, \ldots, c_n, d_1, d_2, \ldots, d_n$$ be integers.

Then $$\prod_{j=1}^n \left({a_j^2 + b_j^2 + c_j^2 + d_j^2}\right) = w^2 + x^2 + y^2 + z^2$$, where $$w, x, y, z \in \Z$$.

What this says is that the product of any number of sums of four squares is also a sum of four squares.

Proof
Taking each of the squares on the RHS and multiplying them out in turn:

$$ $$ $$

$$ $$ $$

$$ $$ $$

$$ $$ $$

All the non-square terms cancel out with each other, leaving:

$$ $$ $$ $$ $$ $$ $$ $$ $$

Proof of Corollary
Follows by induction from the main result.