Solution
= Solution
The linear map
$$
\varepsilon_1\mapsto\varepsilon_1,
\qquad
\varepsilon_2\mapsto\varepsilon_2,
\qquad
\varepsilon_3\mapsto-\varepsilon_3
$$
preserves $\Phi$ and exchanges $\alpha_2$ with $\alpha_3$ while fixing $\alpha_1$. It is not in the $D_3$ <Weyl group>, whose signed permutations change an even number of signs. Thus it is an outer automorphism of the root system.