6(x_1^2 + x_2^2 + x_3^2 + x_4^2)^2 = (x_1 + x_2)^4 + (x_1 + x_3)^4 + (x_2 + x_3)^4 + (x_1 + x_4)^4 + (x_2 + x_4)^4 + (x_3 + x_4)^4 + (x_1 - x_2)^4 + (x_1 - x_3)^4 + (x_2 - x_3)^4 + (x_1 - x_4)^4 + (x_2 - x_4)^4 + (x_3 - x_4)^4. This is proven in Rademacher and Toeplitz.
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