TITLE: Does -I belong to Weyl group?
QUESTION [6 upvotes]: Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$.
If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see if the (unique) longest element in $W$ is $-I$?

REPLY [7 votes]: The mnemonic I use: if the diagram has a natural involution, then $-w_0$ induces it, otherwise $w_0 = -1$. The only place this fails is in $D_n$, where one can switch the antlers, but shouldn't always, so how to remember when? In $D_4$, there's no natural involution, so that helps me remember that in $D_{2n}$, the involution is trivial.