Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}}=\mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $\gamma$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{$\mathcal{P}$} = $\text{Or}$(\gamma)($\mathcal{P}$)$preservesthespaceofPoissonstructuresif$\gamma$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles ${\mathit{\gamma }}_{2\mathrm{\ell }+1}$ is known to exist: marked by $\mathrm{\ell }\in \mathbb{N}$, each of them contains a $(2\mathrm{\ell }+1)$-gon wheel with a nonzero coefficient. At $\mathrm{\ell }=1$ the tetrahedron ${\mathit{\gamma }}_3$ itself is a cocycle; at $\mathrm{\ell }=2$ the Kontsevich–Willwacher pentagon-wheel cocycle ${\mathit{\gamma }}_5$ consists of two graphs. We reconstruct the symmetry $$\mathcal{Q}$_5($\mathcal{P}$) = $\text{Or}$(\boldsymbol{\gamma}_5)($\mathcal{P}$)$andverifythat$$\mathcal{Q}$_5$isaPoissoncocycleindeed:$[![$\mathcal{P}$,$\mathcal{Q}$_5($\mathcal{P}$)]!]\doteq 0$via$[![$\mathcal{P}$,$\mathcal{P}$]!]=0$.