Hi!

Given two Poisson tensors $\pi_0$ and $\pi$ (on a Lie group $G$) ; if the two tensors are compatible, i.e. $$[\pi_0,\pi]=0$$ what are the relations between their hamiltonians ?

If we denote by $X_f$ and $H_f$ the hamiltonians of $f\in C^\infty(G)$, with respect to $\pi_0$ and $\pi$, what can we say about the lie bracket $[X_f,H_f]$ ?

All what I managed to do is to use the graded Jacobi identity, for the Schouten bracket, to show that $$\mathscr{L}_{H_f}\pi_0=\mathscr{L}_{X_f}\pi$$

Any suggestions?