In Schur's inequality When $t=0,2,4\dots$ is an even integer, Wlog set $z=1$, $$x^t(x-y)(x-1)+y^t(y-1)(y-x)+(1-x)(1-y)\ge0$$ how to prove the above inequality is true for all $x,y\in\mathbb{R}$?
(The proof in the link only applies to non-negative $x,y,z$, but here $x,y,z$ are possibly negative)
For $t=0$ it is $\frac{1}{4}(2 x-y-1)^2+\frac{3}{4}(y-1)^2\ge0$.
For $t=2$, I had an ad-hoc proof by sum-of-square method: $$\small x^2(x-y)(x-1)+y^2(y-1)(y-x)+(1-x)(1-y)=\left(\frac{\sqrt{3}}{2} (y-1) (x-y-1)\right)^2+\left(x^2-\frac{x y}{2}-\frac{x}{2}-\frac{y^2}{2}+y-\frac{1}{2}\right)^2$$ so it is non-negative.
For $t\ge4$, I don't know how to prove. For $t=4$ WolframAlpha verifies it is true:
