Remark 1

Let $E,E^*$ be as in Prop. 11, and $\sigma \in$ Aut($\mathfrak{g},\mathfrak{h}$) be such that $\varepsilon (\sigma ) =-w_0$ in the notations of §5, no. 1 (§5, no. 2, Prop. 2). Let $\rho , \rho '$ be the representations of $\mathfrak{g}$ associated to $E,E^*$. Then $\rho \circ \sigma$ is a simple representation of $\mathfrak{g}$ with highest weight $-w_0(\omega )$, so $\rho \circ \sigma$ is equivalent to $\rho '$.