Rewriting xyz as combination of squares

Question

It is well-known that $xy=\frac{1}{4}[(x+y)^2-(x-y)^2]$. However for the problem I am currently working, I need to write $xyz$ as a lineal combination of squares or cubes of $\{\pm x\pm y \pm z\}$.

Any ideas on how to do so?

Edit: As it was pointed to me in the comments, you can't do it with just cuadratics, but I'm still interested in knowing how this might be done with the addition of cubes

Answer 1

$$ x y z = \frac{1}{24}\left( (x+y+z)^3 - (x+y-z)^3 -(x-y+z)^3 -(-x+y+z)^3 \right) \text{.} $$