Consider a determinant of a Gram matrix in dimension $4$.
$$\begin{vmatrix} 1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\ -\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\cos(\alpha_5)\\ -\cos(\alpha_2) & -\cos(\alpha_6) & 1& -\cos(\alpha_4)\\ -\cos(\alpha_3) & -\cos(\alpha_5) & -\cos(\alpha_4)& 1\\ \end{vmatrix}$$
and expand it by opening all brackets and using the identity
$$\cos(\alpha_i)=\dfrac{e^{\alpha_i}+e^{-\alpha_i}}{2}.$$
One gets the following answer: \begin{equation} G_T=-\frac{1}{8}\sum_{r\cdot e_7=0} (-1)^{r\cdot e_8}\chi(r)+\frac{1}{16}\sum_{\substack{r \cdot e_8 \neq 0\\ r\cdot e_7\neq 0}}\chi(2r)-1. \end{equation} Here $r$ goes along all roots of $E_8,$ $e_1, e_2, \ldots, e_8$ are some pairwise orthogonal roots of $E_8$ and $\chi$ is a character, defined by the rule $\chi(2e_i)=e^{2i\alpha_i}$ for $i\leq 6$, $\chi(2e_7)=\chi(2e_8)=1.$ From this expression one immediately sees that the Gram matrix determinant does not change under Regge symmetries (see this question Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra).
Question Is there a Lie algebra explanation of that? Do there exist any similar identities for other root systems?
