By equation $H^{(L)}(\partial)(z,w)$ has no negative powers of $z$, and by Lemma \[20130925:lem1\](a) it has no negative powers of $w$.

Adler type scalar pseudodifferential operators
----------------------------------------------

\[def:adler\] Let ${\mathcal{V}}$ be a differential algebra endowed with a $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}$. We call a pseudodifferential operator $L(\partial)\in{\mathcal{V}}((\partial^{-1}))$ of *Adler type* (for the $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}$) if the following identity holds in ${\mathcal{V}}[\lambda]((z^{-1},w^{-1}))$: $$\label{generating}
\{L(z)_\lambda L(w)\}=H^{(L)}(\lambda)(w,z)
\,.$$

\[20131027:prop1\] Let ${\mathcal{V}}$ be a differential algebra, let $\{\cdot\,_\lambda\,\cdot\}$ be a $\lambda$-bracket on ${\mathcal{V}}$, and let $L(\partial)\in{\mathcal{V}}((\partial^{-1}))$ be an Adler type pseudodifferential operator. Then:

(a) The following identity holds in ${\mathcal{V}}[\lambda]((z^{-1},w^{-1}))$: $$\label{skew-symseries}
    \{L(z)_{\lambda}L(w)\}=-\{L(w)_{-\lambda-\partial}L(z)\}\,.$$

(b) The following identity holds in ${\mathcal{V}}[\lambda,\mu]((z_1^{-1},z_2^{-1},z_3^{-1}))$: $$\label{jacobiseries}
    \{L(z_1)_{\lambda}\{L(z_2)_{\mu}L(z_3)\}\}
    -\{L(z_2)_{\mu}\{L(z_1)_{\lambda}L(z_3)\}\}
    =\{\{L(z_1)_{\lambda}L(z_2)\}_{\lambda+\mu}L(z_3)\}\,.$$

In view of equation , part (a) follows from Lemma \[20130925:lem1\](a). Let us prove part (b). Using sesquilinearity, the left and right Leibniz rules and equation we can rewrite each term of the equation as follows. The first one is $$\begin{aligned}
&\{L(z_1)_{\lambda}\{L(z_2)_{\mu}L(z_3)\}\}\notag\\
&=H^{(L)}(\lambda)(z_2,z_1)
i_{z_2}(z_2-z_3-\mu-\partial)^{-1}
L(z_3)
\label{eq:a}\\
&+L(z_2)
i_{z_2}(z_2-z_3-\lambda-\mu-\partial)^{-1}
H^{(L)}(\lambda)(z_3,z_1)
\label{eq:b}\\
&-H^{(L)}(\lambda)(z_3+\mu+\partial,z_1)
i_{z_2}(z_2-z_3-\mu-\partial)^{-1}
L^*(-z_2+\mu)
\label{eq:c}\\
&-L(z_3+\lambda+\mu+\partial)
i_{z_2}(z_2-z_3-\lambda-\mu-\partial)^{-1}
\Big(\Big|_{x=\lambda+\mu+\partial}
\!\!\!
H^{(L)}(\lambda)(z_2-x,z_1)
\Big)
\,;\label{eq:d}\end{aligned}$$ the second term is $$\begin{aligned}
&\{L(z_2)_{\mu}\{L(z_1)_{\lambda}L(z_3)\}_{H}\}_{H}\notag\\
&=H^{(L)}(\mu)(z_1,z_2)
i_{z_1}(z_1-z_3-\lambda-\partial)^{-1}
L(z_3)
\label{eq:a'}\\
&+L(z_1)
i_{z_1}(z_1-z_3-\lambda-\mu-\partial)^{-1}
H^{(L)}(\mu)(z_3,z_2)
\label{eq:b'}\\
&-H^{(L)}(\mu)(z_3+\lambda+\partial,z_2)
i_{z_1}(z_1-z_3-\lambda-\partial)^{-1}
L^*(-z_1+\lambda)
\label{eq:c'}\\
&-L(z_3+\lambda+\mu+\partial)
i_{z_1}(z_1-z_3-\lambda-\mu-\partial)^{-1}
\Big(\Big|_{x=\lambda+\mu+\partial}
\!\!\!
H^{(L)}(\mu)(z_1-x,z_2)
\Big)
\,;
\label{eq:d'}\end{aligned}$$ the third term is $$\begin{aligned}
&\{\{L(z_1)_{\lambda}L(z_2)\}_{H}{}_{\lambda+\mu}L(z_3)\}_{H}
\notag\\
