Exercise 12, § 1

Let $R$ be a root system in $V$ of rank $\geq 3$, $(\alpha_1, \ldots, \alpha_l)$ a basis of $R$. $V'$ (resp. $V''$) the subspace of $V$ generated by the $\alpha_i$ for $i \geq 2$ (resp. $i \geq 3$). $R' = R \cap V'$, $R'' = R \cap V''$. Let $d$ (resp. $d', d''$) be the determinant of the Cartan matrix of $R$ (resp. $R', R''$). Assume that $\alpha_1$ is orthogonal to all the $\alpha_i$ except $\alpha_2$, and that $\| \alpha_1 \| = \| \alpha_2 \|$. Show that $d = 2d' - d''$.