Consider a Hamiltonian system $\dot{x} = X_H(x) + X_C(x) u$ and $y = C(x)$ where $u$ is a scalar and $x = (q,p) \in \mathbb R^{2n}$ and $H$ and $C$ are some functions mapping $\mathbb R^{2n} \to \mathbb R$. The vector field $X_F$ is defined as $\dot{q_i} = F_{p_i}$ and $\dot{p_i} = -F_{q_i}$.
My professor claimed that there exists an isomorphism between the {strong accessibility algebra under the Lie-bracket $[\cdot,\cdot]$} and the {observation space under the Poission bracket $\{\cdot, \cdot\}$}.
Question: I am struggling to find this isomorphism however. Can anyone help?
My attempt: The strong accessibility algebra is given by $$\mathcal{C}_{sa} = \operatorname{span}\{ X_C, [X_H,X_C], [X_H,[X_H,X_C]],[X_C,[X_H,X_C]], \text{ higher order Lie-brackets} \}, $$ while the observation space is defined as $$ \mathcal{O}_s = \operatorname{span} \{ C, L_{X_H}C,L_{X_C}L_{X_H}C, L_{X_C}L_{X_H}C, \text{higher order Lie-derivatives} \}$$ Now the isomorphism seems obvious. Let $\phi : \mathcal{C}_{sa} \to \mathcal{O}_s$ via $$ \phi([X_1,[X_2,\ldots ,[X_H, X_C]\ldots]) = L_{X_1}L_{X_2} \ldots L_{X_H} C $$ where $X_i \in \{ X_H, X_C\}$. But I do not know how to show that this is an isomorphism. Thanks in advance for any help!