could any one tell me why not all automorphisms of a root system are elements of weyl group? For example in $A_n, n>2$ the automorphism $\alpha\mapsto -\alpha$ is not in the weyl group. I do not understand why? it is also a reflection and must be an element of the weyl group. thank you. well $A_n$ means root systm of rank $n$.
not all automorphisms of a root system are elements of weyl group
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$\begingroup$
lie-algebras
root-systems