Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to D.$$ I need an explicit formula for a bimodule splitting of this map or equivalently an element $z \in D\otimes D$ s.t. $az=za$ for any $a \in D$ and $m(z)=1$.

It is possible to use isomorphism of algebras $Cl(V^* \oplus V) \cong End(\wedge V)$ and for $End(\wedge V)$ such splitting is given (up to sign) by the same formula as for matrix algebra. So, I know that such splitting exists and I want a nice formula in term of differential operators (or standard generators of Clifford algebra).


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