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Root systems and sums of squares

It is easy to see that the quadratic form for the root system $A_n$ is a sum of $n+1$ squares of integral linear forms: $$q_{A_n} = 2 \sum_{i=1}^n x_i^2 - 2 \sum_{i=1}^{n-1} x_i x_{i+1} = x_1^2 + ...