value zero modulo local terms). As mentioned above, differential operators of positive normal order will be left out from the commutator considerations. So will operators of type $P^{(3)}$, since the structure is not preserved under differentiation of symbols.

At this point we can state a part of the results we will show in this paper:

Let $A=P_++G$ and $A'=P'_++G'$ be of orders $\nu $ resp. $\nu
'$, with $P$ resp. $P'$ vanishing if $\nu $ resp. $\nu ' \notin \Bbb
Z$, and with $G$ and $G'$ of class $0$. Let $P_1$ and $P_2$ be two auxiliary operators as explained above. Then

[(i)]{} $$C_{0}(A,P_{1,\operatorname{D}})-C_{0}(A,P_{2,\operatorname{D}})\text{
is locally determined};\tag2.7$$ it vanishes if $\nu <-n$ or $\nu \notin \Bbb Z$, and otherwise depends solely on the terms of the first $\nu +n+1$ homogeneity degrees in the symbols of $P_++G $, $P_1$ and $P_2$.

[(ii)]{} If $\nu $ and $\nu '\in\Bbb Z$, assume that $P$ and $P'$ have normal order $0$; if $\nu $ or $\nu '\in \Bbb R\setminus \Bbb Z
$, assume that $P$ and $P'$ are zero. Then $$C_{0}([A,A'],P_{1,\operatorname{D}})\text{
is locally determined};\tag2.8$$ it vanishes if $\nu +\nu '<-n$ or $\nu +\nu '\notin \Bbb Z$, and otherwise depends solely on the terms of the 