A polynomial identity

Expansion.
$$(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=a^3+ab^2+ac^2-a^2b-abc-a^2c+ba^2+b^3+bc^2-ab^2-b^2c-abc+ca^2+cb^2+c^3-abc-ac^2-bc^2.$$
Group: $a^3+b^3+c^3$, and for each mixed term, we have:
- $ab^2$ terms: $ab^2-ab^2=0$.
- Similarly for others.

After cancellation: $a^3+b^3+c^3-3abc$.

Conclusion. The identity holds.

Corollary. $a^3+b^3+c^3=3abc$ iff $a+b+c=0$ or $a=b=c$ (over $\mathbb{R}$, using the SOS identity for the second factor).