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 $-I$?



Does -I belong to Weyl group?
As Koen S points out, the longest element of an irreducible Weyl group is treated in an earlier question (in fact, it comes up in several questions). The question asked here presupposes a standard linear realization of the Weyl group, as occurs in the structure theory of a semisimple Lie algebra over $\mathbb{C}$ for instance. In this context, the standard answer given by Max is formulated as Exercise 5 at the end of Section 13 in my 1972 Springer graduate text. However one arrives at this conclusion, it obviously depends on the classification of irreducible root systems.


On the other hand, the question makes sense for any irreducible finite Coxeter group in its usual realization as a reflection group, and is approached in this spirit (via the Coxeter element) toward the end of Section 3.19 in my 1990 book on reflection groups and Coxeter groups. 


P.S. For irreducible Weyl groups, a natural motivation for asking this question involves the criterion for all finite dimensional irreducible representations of the associated simple Lie algebra to be *self-dual*: If the given highest weight is $\lambda$, the dual has highest weight $-w\_0 \lambda$ (where $w\_0$ is the longest element of $W$). This always coincides with $\lambda$ iff $-1 \in W$. 


