<s>[INST] How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong?
$$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$
Thanks
 [/INST] This is the two-variable identity $(X+Y)^3 - (X^3 + Y^3) = 3XY(X+Y)$  
presented as a formula for $X^3 + Y^3 + Z^3$ when $X+Y+Z=0$  (by setting $Z = -(X+Y)$),   
and then using $(X,Y,Z)=(a-b,b-c,c-a)$ to parametrize solutions of $X+Y+Z=0$.
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