Integer as Sum of 4 Cubes

Theorem
Let $n \in \Z$ be an integer.

Let $n \not \equiv 4 \pmod 9$ and $n \not \equiv 5 \pmod 9$.

Then it is possible to express $n$ as the sum of no more than $4$ cubes which may be either positive or negative.

Also see

 * Compare with the Hilbert-Waring theorem for $k = 3$: if the cubes all have to be positive then as many as $9$ may be needed.