Let $(W,S)$ be a finite, irreducible Coxeter Group. I thought it was true (from Humphrey's book Ex 2 on page 82) that if the Coxeter Number of $W$, $h$, is even then $$c^{h/2} = \omega_0 \quad \dagger$$ is true for any Coxeter Element $c$, and $\omega_0$ is the longest element of $W$.
Is this true and have I misunderstood what this exercise is saying?
When I was playing with the $D_5$ Coxeter Group, I found that some, but not all, Coxeter elements have satisfy $\dagger$. Specifically, of all permutations of the 5 generators exactly 2/3s, had their associated Coxeter Element satisfy $\dagger$.
Can someone help point out what my misunderstanding is?
Thanks,
Rob