Gram matrix determinant in dimension 4 and $E_8$

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: $$$$G_T=-\frac{1}{8}\sum_{r\cdot e_7=0} (-1)^{r\cdot e_8}\chi(r)+\frac{1}{16}\sum_{\substack{r \cdot e_8 \neq 0\\ r\cdot e_7\neq 0}}\chi(2r)-1.$$$$ Here $$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?

• What are $e_L$ and $e_A$? – Vít Tuček Mar 26 '19 at 21:38
• Thanks for noticing, this is a typo. – Daniil Rudenko Mar 31 '19 at 11:14