Let the symmetric group $S_4$ act on $\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$ by permuting the 1st $4$ variables and again permuting the last $4$ variables. We can restrict the action to the alternating group $A_4$. It seems there is a degree $4$ polynomial which is invariant under the $A_4$ action but not invariant under the $S_4$ action. But I can't figure out what is that polynomial. Any help in this direction will be helpful.