An example concerning Hamiltonian groups of self product, II

We describe the natural identification of $FH_*(X \times X, \triangle; \omega \oplus -\omega)$ with $FH_*(X, \omega)$. Under this identification, we show that the extra elements in $Ham(X \times X, \omega \oplus -\omega)$ found in (Part I), for $X = (S^2 \times S^2, \omega_0 \oplus \lambda \omega_0)$ for $\lambda > 1$, do not define new invertible elements in $FH_*(X, \omega)$.