Remark 1

Let $\lambda \in P_{++},\rho_{\lambda}$ the representation of $\mathfrak{g}$ on $E(\lambda ),s\in$ Aut($\mathfrak{g}$), and $\sigma$ the canonical image of $s$ in Aut(R$,B$) $($§5, no. 3, Cor. 1 of Prop. 5). Then $\rho_{\lambda}\circ s$ is equivalent to $\rho_{\sigma \lambda}$; indeed, if $s\in$ Aut$_0(\mathfrak{g}),\rho_{\lambda}\circ s$ and $\rho_{\sigma \lambda}$ are equivalent to $\rho_{\lambda}$ (Prop. 2); and, if $s$ leaves $\mathfrak{h}$ and B stable, $\rho_{\lambda}\circ s$ is simple of highest weight $\sigma \lambda$.

In particular, the fundamental representations are permuted by $s$, and this permutation is the identity if and only if $s\in$ Aut$_0(\mathfrak{g})$.