Bott and Tu claim that a sphere bundle $E$ cohomology whenever a global closed n-form exists can be decomposed as
$$H^*(E)=H^*(M) \otimes H^*(S^n)$$
which is fine except that they claim this follows from Leray-Hirsch. But that theorem on pp 50 states this composition exists if there are global forms which generate the cohomology of the fiber not just an n-form. Does the existence of a closed n-form in this case imply there are global forms which generate the cohomology of the fiber each time?