Symmetric combinations of variables with omega (part 3)

If $1$, $\omega$, ${\omega }^2$ are the three cube roots of unity, and $x=a+b$, $y=a\omega +b{\omega }^2$, $z=a{\omega }^2+b\omega$, shew that $x^3+y^3+z^3=3(a^3+b^3)$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VIII. Surds and Imaginary Quantities, Examples VIII.b, No. 30 (part 3)