sequence $$0 \to \Psi^{-\infty}(M,E) \to \Psi_H(M,E) \to {\mathcal{S}}_H(M,E) \to 0$$ and similarly for the algebra of compactly-supported operators $\Psi_{H,c}(M,E)$. Note that for $a \in {\mathcal{S}}_H(M,E)$, $(x,p) \in T^{*}_xM$, we have $a(x,p) \in {\mathrm{End}}(E_x)$. Let ${\mathcal{PS}}_H(M,E) \subset {\mathcal{S}}_H(M,E)$ denote the subalgebra of polynomial Heisenberg symbols (with respect to the cotangent coordinate $p$). The latter is isomorphic to the algebra of differential operators, endowed with the Heisenberg degree.\
The vector bundle of interest in this paper will be the exterior algebra $E=\Lambda^{\bullet} (T^*M\otimes{\mathbb{C}})$ of the complexified cotangent bundle. In a distinguish coordinate system $(x^{1}, \ldots, x^{n})$ over an open subset $U \subset M$, a local basis of the sections of $E$ is given by $ 1,dx^{i_1}, dx^{i_1} \wedge dx^{i_2}, \ldots, dx^{i_1} \wedge \ldots \wedge dx^{i_n}$, $1 \leq i_1< \ldots <i_n \leq n $. Moreover, the endomorphisms ${\mathrm{End}}(E_x)$ of the fibre $E_x$ are generated by $$\psi^{i} = dx^{i} \wedge \textbf{.}, \quad \overline{\psi}_i = \iota(\partial_{x^{i}})$$ for $i = 1, \ldots, n$, where $\iota$ stands for the interior product with a vector field. We have the following *anti-commutation relations rules* : $$[\psi^{i}, \overline{\psi}_j] = \delta_i^{j}, \quad [\psi^{i}, \psi^{j}] = [\overline{\psi}_i, \overline{\psi}_j] = 0$$ In other words, ${\mathrm{End}}(E_x)$ is the Clifford algebra of $T_x M \oplus 