Let a d = b c, then 64[(a + b + c)^6 + (b + c + d)^6 - (c + d + a)^6 - (d + a + b)^6 + (a - d)^6 - (b - c)^6][(a + b + c)^10 + (b + c + d)^10 - (c + d + a)^10 - (d + a + b)^10 + (a - d)^10 - (b - c)^10] = 45[(a + b + c)^8 + (b + c + d)^8 - (c + d + a)^8 - (d + a + b)^8 + (a - d)^8 - (b - c)^8]^2. This can also be expressed by defining F_(2m)(a, b, c, d) = (a + b + c)^(2m) + (b + c + d)^(2m) - (c + d + a)^(2m) - (d + a + b)^(2m) + (a - d)^(2m) - (b - c)^(2m) f_(2m)(x, y) = (1 + x + y)^(2m) + (x + y + x y)^(2m) - (y + x y + 1)^(2m) - (x y + 1 + x)^(2m) + (1 - x y)^(2m) - (x - y)^(2m).

Eisenstein integer | Hirschhorn 3-7-5 identity

Srinivasa Aiyangar Ramanujan