Prerequisites : (1) : a^3+b^3=(a+b)^3-3ab(a+b)............(ast).
(2) : c^3+d^3=(c+d)(c^2-cd+d^2)..........................(astast).
"The Expression"=ul(x^3+y^3)+z^3-3xyz,
=ul{(x+y)^3-3xy(x+y)}+z^3-3xyz............[because, (ast)],
=(x+y)^3+z^3-ul(3xy(x+y)-3xyx),
=ul(u^3+z^3)-ul(3xyu-3xyz)," say, where, "u=(x+y),
=ul((u+z))(u^2-uz+z^2)-3xyul((u+z))......[because, (astast)],
=(u+z)(u^2+z^2-uz-3xy).
Now reverting from u" to "(x+y), we get,
"The Exp."={(x+y)+z}{(x+y)^2+z^2-(x+y)z-3xy},
=(x+y+z){(x^2+ul(2xy)+y^2)+z^2-ul(3xy)-(x+y)z},
=(x+y+z)(x^2+y^2+z^2-xy-yz-zx),
=1/2(x+y+z)(2x^2+2y^2+2z^2-2xy-2yz-2zx),
=1/2(x+y+z){(x^2-2xy+y^2)+(y^2-2yz+z^2)+(z^2-2zx+x^2)},
=1/2(x+y+z){(x-y)^2+(y-z)^2+(z-x)^2}, as desired!