Prove

we know that (1+w+w²) = 0 and w³ =1

x² + y² +z² = a²+b²+2ab+a²w²+b²w⁴+2abw³+a²w⁴+b²w²+2abw³

now putting w³=1 and w⁴= w

 x² + y² +z² = a² (1+w+w²)+b² (1+w+w²)+6ab= 6ab