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This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question.

The following fact could be extracted from 0402087:

For any $a_i\neq 0$ a polynomial

$$(t+a_1a_2a_3)(t+a_2a_4a_6)(t+a_1a_5a_6)(t+a_3a_4a_5)-$$

$$(t-a_1a_2a_4a_5)(t-a_1a_3a_4a_6)(t-a_2a_3a_5a_6)(t-1)$$

has a nonzero double root if and only if Gram determinant

$$ \begin{vmatrix} 1 & -\frac{a_1+\frac{1}{a_1}}{2}& -\frac{a_2+\frac{1}{a_2}}{2}&-\frac{a_3+\frac{1}{a_3}}{2} \\ -\frac{a_1+\frac{1}{a_1}}{2} & 1 &-\frac{a_6+\frac{1}{a_6}}{2}&-\frac{a_5+\frac{1}{a_5}}{2}\\ -\frac{a_2+\frac{1}{a_2}}{2} & -\frac{a_6+\frac{1}{a_6}}{2} & 1&-\frac{a_4+\frac{1}{a_4}}{2}\\ -\frac{a_3+\frac{1}{a_3}}{2} & -\frac{a_5+\frac{1}{a_5}}{2} & -\frac{a_4+\frac{1}{a_4}}{2} & 1\\ \end{vmatrix} $$

vanishes.

Question: How to prove it? I was able only to verify it in Maple.

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