An example concerning Hamiltonian groups of self product, I

We show that $(S^2\times S^2, \omega_0 \oplus \lambda\omega_0)$, with $\lambda > 1$, is an example of symplectic manifold $(X, \omega)$ such that the $\pi_1 Ham(X \times X, \omega\oplus -\omega)$ contains extra elements than those from $\pi_1 Ham(X, \omega) \times \pi_1 Ham(X, -\omega)$.