we shall use the following notation: $$V_{\lambda,\mu}:=V[[\lambda^{-1},\mu^{-1},(\lambda+\mu)^{-1}]][\lambda,\mu]\,,$$ namely, the quotient of the ${\mathbb{F}}[\lambda,\mu,\nu]$-module $V[[\lambda^{-1},\mu^{-1},\nu^{-1}]][\lambda,\mu,\nu]$ by the submodule $(\nu-\lambda-\mu)V[[\lambda^{-1},\mu^{-1},\nu^{-1}]][\lambda,\mu,\nu]$. We have the natural embedding $\iota_{\mu,\lambda}:\,V_{\lambda,\mu}\hookrightarrow V((\lambda^{-1}))((\mu^{-1}))$ defined by expanding the negative powers of $\nu=\lambda+\mu$ by geometric series in the domain $|\mu|>|\lambda|$.

Let ${\mathcal{V}}$ be a differential algebra. A *non-local* $\lambda$-*bracket* on ${\mathcal{V}}$ is an ${\mathbb{F}}$-linear map $\{\cdot\,_\lambda\,\cdot\}:\,{\mathcal{V}}\otimes {\mathcal{V}}\to {\mathcal{V}}((\lambda^{-1}))$ satisfying the sesquilinearity conditions and the Leibniz rules, as in Definition \[def:lambda\]. It is called *skew-symmetric* if the skew-symmetry condition in Definition \[def:pva\] holds as well. The term $\{g_{-\lambda-\partial}f\}$ in the RHS of the skewsymmetry condition should be interpreted as follows: we move $-\lambda-\partial$ to the left and we expand it in non-negative powers of $\partial$, acting on the coefficients of the $\lambda$-bracket. Clearly, from skew-symmetry and the left Leibniz rule, we also have the right Leibniz rule, which should be interpreted in a similar way.

The non-local skew-symmetric $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}$ is called *admissible* if $$\{f_\lambda\{g_\mu h\}\}\in{\mathcal{V}}_{\lambda,\mu}
\qquad
\text{ for all } f,g,h\in{\mathcal{V}}\,.
$$ Here we are identifying the space ${\mathcal{V}}_{\lambda,\mu}$ with its image in ${\mathcal{V}}((\lambda^{-1}))((\mu^{-1}))$ via the embedding $\iota_{\mu,\lambda}$. Note that, from skew-symmetry, we also have that $\{g_\mu\{f_\lambda h\}\}\in{\mathcal{V}}_{\lambda,\mu}$ and $\{\{f_\lambda g\}_{\lambda+\mu} h\}\in{\mathcal{V}}_{\lambda,\mu}$. Therefore, the Jacobi identity can be understood as 