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An automorphism of a root system outside the Weyl group is the sign changeIt preserves the set $\{\pm {\epsilon }_i\pm {\epsilon }_j\}$, fixes ${\alpha }_1$, and interchanges ${\alpha }_2$ and ${\alpha }_3$. It therefore realizes the nontrivial symmetry of the Dynkin diagram while preserving the chosen root basis.

To verify that $\tau \notin W$, note that $s_1$ swaps ${\epsilon }_1,{\epsilon }_2$, $s_2$ swaps ${\epsilon }_2,{\epsilon }_3$, and $s_3$ swaps ${\epsilon }_2,{\epsilon }_3$ while changing both their signs. Every product of these generators is a signed permutation with an even number of sign changes, whereas $\tau$ has one sign change. Thus it cannot be a Weyl group element. The parity here concerns sign changes, rather than the determinant of the permutation.