Remark 2

Assume that $w_0=-1$. Then, for any finite dimensional $\mathfrak{g}$-module E, E is isomorphic to $E^*$. Recall that, if $\mathfrak{g}$ is simple, $w_0=-1$ in the following cases: $\mathfrak{g}$ of type $A_1,B_l(l\geq 2), C_l(l\geq 2), D_l(l$ even $\geq 4), E_7,E_8,F_4,G_2$ (Chap. VI, Plates).