I am trying to solve exercise 10.12 from humphreys lie algebra book. I need to prove that if an element $\sigma$ of the Weyl group is such that $\sigma v=v$ for $v$ a vector in the fundamental Weyl chamber then $\sigma$ must be the identity.
My attempt was trying to prove that $\sigma $ fixes all vectors in the weyl chamber which would imply that $\sigma$ fixes a basis of $E$ an therefore is the identity. Unfortunately I couldn't prove this so I don't know what to do.