Exercise 7, § 1

The connection index of a root system is equal to the determinant of its Cartan matrix. (Use formula (14) of no. 10. Show on the other hand that, in a real vector space with a scalar product, if $(\varepsilon_1, \ldots, \varepsilon_n), (\varepsilon'_1, \ldots, \varepsilon'_n)$ are two bases such that $(\varepsilon_i|\varepsilon'_j) = \delta_{ij}$, then $\det_{(\varepsilon_1, \ldots, \varepsilon_n)}(\varepsilon'_1, \ldots, \varepsilon'_n) > 0.$)