Proving $2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)=3(a^2+b^2+c^2+(a+b+c)^2)^2$ in another way? How do I prove the following identity without expanding both sides directly.
$$2((a+b)^4+(a+c)^4+(b+c)^4)+4(a^4+b^4+c^4+(a+b+c)^4)\\=3(a^2+b^2+c^2+(a+b+c)^2)^2$$
I expanded both sides directly and it is true. However, I was hoping there could be another way to prove it, like that of Candido's identity which could be proved using diagrams.

Why do this by hand?
Mathematica:
Simplify[2 ((a + b)^4 + (a + c)^4 + (b + c)^4) +  4 (a^4 + b^4 + c^4 + (a + b + c)^4) ==  3 (a^2 + b^2 + c^2 + (a + b + c)^2)^2]
(* True *)
