# Action of Weyl group on the dual Cartan subalgebra in bad characteristic

I am interested in an analogue of Proposition 14.31 in Fulton and Harris' book 'Representation Theory: A First Course' for positive characteristic. Let $G$ be a semisimple algebraic group of adjoint type defined over an algebraically closed field $K$ of characteristic 2, and $\mathfrak{g} = Lie(G)$. Suppose that $\mathfrak{g}$ is a Lie algebra of classical type $B, C$ or $D$. Let $T$ a maximal torus of $G$, and $W = N(T)/T$ the Weyl group of $T$. Set $\mathfrak{h} = Lie(T)$, a Cartan subalgebra of $\mathfrak{g}$, and let $\mathfrak{h}^*$ denote the dual vector space of $\mathfrak{h}$. Since I am working in bad characteristic, I don't have a $G$-equivariant isomorphism $\mathfrak{h} \to \mathfrak{h}^*$.

Now Fulton and Harris state that, in characteristic 0, the Weyl group $W$ acts irreducibly on $\mathfrak{h}^*$. Their proof requires that $\mathfrak{g}$ is a simple Lie algebra, which is not true in general over fields of characteristic 2, and that such a $G$-equivariant isomorphism $\mathfrak{h} \to \mathfrak{h}^*$ exists. It's easy enough to see that their proof works for fields of positive characteristic $p > 2$, since in this case the characteristic of $K$ is very good for $G$ and one can just apply the given proof in this situation.

Is it true that, when the characteristic of $K$ is 2, $W$ has an irreducible action on $\mathfrak{h}^*$, or does this property fail? Any reference or counterexample would be appreciated.

It fails very often! For instance, the Weyl group $S_3$ of type $A_2$ in characteristic $3$ has only two irreducible representations, both of dimension $1$. So there is no chance for it to act irreducibly on a two-dimensional vector space. (That is, it's not true that $W$ acts irreducibly on the dual Cartan in characteristic $p>2$).
You can see explicitly that the reflection representation of $S_3$ is reducible, as it contains the $S_3$-fixed element
$$e_1+e_2+e_3=e_1-2e_2+e_3$$
Its dual then has the trivial module as a quotient. The easiest characteristic two example is probably the reflection representation of $W(A_3)=S_4$ or that of $W(B_2)$, both of which are reducible in characteristic $2$.