Question 6
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$
P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) .
$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$
P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) .
$$
$$
P(x, y, z)=P(x, y, x y-z)=P(x, z x-y, z)=P(y z-x, y, z) .
$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$
P(x, y, z)=F\left(x^{2}+y^{2}+z^{2}-x y z\right) .
$$