Exercise 5, § 4

Let $R$ be a root system, and $P'(R)$ the subgroup of $P(R)$ introduced in $§ 1$, Exerc. 8. Show that $P'(R) = P(R)$ if $R$ is of type $A_l$ with $l$ even, or $B_l$ with $l \equiv 0, 3 \pmod{4}$, or $D_l$ with $l \equiv 0, 1 \pmod{4}$, or $G_2$, or $F_4$, or $E_6$, or $E_8$. Show that $P'(R) = Q(R)$ if $R$ is of type $C_l$, or $B_l$ with $l \equiv 1, 2 \pmod{4}$, or $E_7$, or $A_1$. If $R$ is of type $A_l$ with $l$ odd and $> 1$, $P'(R)/Q(R)$ is the unique subgroup of index 2 in the cyclic group $P(R)/Q(R)$. If $R$ is of type $D_l$ with $l \equiv 2, 3 \pmod{4}$, $P'(R)/Q(R)$ is the unique subgroup of order 2 of $P(R)/Q(R)$ stable under $\Lambda(R)$.