  Given a pair of (real or complex) Lie algebroid structures on a vector bundle
$A$ (over $M$) and its dual $A^*$, and a line bundle $\module$ such that
$\module\otimes\module=(\wedge^{\TOP} A^*\otimes\wedge^{\TOP} T^*M)$, there
exist two canonically defined differential operators $\bdees$ and $\bdel$ on
$\sections{\wedge A\otimes\module}$. We prove that the pair $(A,A^*)$
constitutes a Lie bialgebroid if, and only if, the square of $\bdirac
=\bdees+\bdel$ is the multiplication by a function on $M$. As a consequence, we
obtain that the pair $(A,A^*)$ is a Lie bialgebroid if, and only if, $\bdirac$
is a Dirac generating operator as defined by Alekseev & Xu \cite{AlekseevXu}.
Our approach is to establish a list of new identities relating the Lie
algebroid structures on $A$ and $A^*$ (Theorem \ref{Thm:C}).
