A Poisson structure on a manifold \(P\) is a bracket \[\{\cdot,\cdot\}:\wedge^2 C^\infty(P)\to C^\infty(P)\] that is a Lie bracket and at the same time a bi-derivation.
\(\{h,\}\) is a derivation on \(C^\infty(P)\) and thus corresponds to a vector field \(\xi_h \in \mathfrak{X}(P)\), called the Hamiltonian vector field associated to \(h\).
Any symplectic manifold is Poisson, the bracket can be given as \(\{f,g\} = \omega(\xi_f,\xi_g) = \xi_f (\mathrm{d} g)\)
By the Jacobian identity,
\[\begin{align*} \xi_{\{f,g\}} h &= \{\{f,g\},h\} \\ &= \{\{f,h\},g\} + \{f,\{g,h\}\} \\ &= -\xi_g(\xi_f h) + \xi_f(\xi_g h) \\ &= [\xi_f,\xi_g] h \end{align*}\]
We have that \(\xi_{\{f,g\}} = [\xi_f,\xi_g]\). Note that if you use the other convention \(\{,f\} = \xi_f\), you get an extra minus sign.
A smooth map \(\varphi:P_1\to P_2\) between two Poisson manifolds \((P_1,\{,\}_{P_1})\) and \((P_2, \{,\}_{P_2})\) is Poisson if \[\varphi^*\{f,g\}_{P_2}=\{\varphi^*f,\varphi^*g\}_{P_1}.\]
A Poisson submanifold \(Q\subset P\) is a submanifold \(Q\) of \(P\) equipped with a Poisson structure that makes the inclusion map Poisson.
For two Poisson manifolds \(P_1,P_2\), we can define the Poisson structure on the product manifold \(P_1\times P_2\) to be the unique Poisson structure such that
The projection maps \(\pi_1:P_1\times P_2\to P_1\) and \(\pi_2:P_1\times P_2\to P_2\) are Poisson maps.
\(\left\{\pi_1^*C^\infty(P_1),\pi_2^*C^\infty(P_2)\right\}_{P_1\times P_2}=0\).
This basically means the vertical functions have no horizontal derivatives and vice versa, making sure that \(P_1\) and \(P_2\) directions are independent.
Let \(P\) be a Poisson manifold with \(p\in P\). Then around \(p\), locally there is an diffeomorphic Poisson map \(\varphi = \varphi_S\times \varphi_N: U_p\to S\times N\) to a product of a symplectic manifold \(S\) and a Poisson manifold \(N\) whose Poisson rank is \(0\) at \(p\).