.$$ In turn, we have $$\label{Eqt:overlineHsharpcircHsharp}
{\overline{H}{^{\sharp}}}{H{^{\sharp}}}=\left(
\begin{matrix}
  {\overline{\theta}{^{\flat}}}\circ\theta{^{\flat}}+ {\overline{\pi}{^{\sharp}}}\circ\omega{^{\flat}}& {\overline{\theta}{^{\flat}}}\circ \pi{^{\sharp}}+{\overline{\pi}{^{\sharp}}}\circ\theta{^{\sharp}}\\
  {\overline{\omega}{^{\flat}}}\circ \theta{^{\flat}}+{\overline{\theta}{^{\sharp}}}\circ
  \omega{^{\flat}}& {\overline{\omega}{^{\flat}}}\circ\pi{^{\sharp}}+
  {\overline{\theta}{^{\sharp}}}\circ \theta{^{\sharp}}\end{matrix} \right).$$

Given an extended Poisson manifold $({\boldsymbol{X}},H)$, let ${A}={{T^{1,0}{\boldsymbol{X}}}}\bowtie{{(T^{0,1}{\boldsymbol{X}})^*}}$. Then ${{A}^*}_H$ is the $(\pm{i})$-eigenbundle of a generalized complex structure if and only if ${\overline{H}{^{\sharp}}}{H{^{\sharp}}}-{\operatorname{id}_{{{A}^*}}}$ is invertible.

If $H=\pi$ (i.e. H is a holomorphic Poisson bivector field) or $H=\omega$, it is clear that ${\overline{H}{^{\sharp}}}{H{^{\sharp}}}$ is zero. Hence, in these two situations, the extended Poisson structure on ${\boldsymbol{X}}$ is actually a generalized complex structure.

Here is a simple example of extended Poisson structure, which does not arise from a generalized complex structure.

\[Ex:realtorus\] Consider the torus $\mathbf{T}={\mathbb{C}}/({\mathbb{Z}}+{i}{\mathbb{Z}})$ with its standard complex structure. Let $z$ be the standard coordinate on $\mathbf{T}$. Obviously, any $$\label{Eq:thetafddzdbarz}
\theta=f(z,\bar{z})\tfrac{d}{dz}\wedge d\bar{z},$$ where $f$ is a smooth ${\mathbb{C}}$-valued function, is an extended Poisson structure. In this case, ${\overline{H}{^{\sharp}}}{H{^{\sharp}}}={\left\vertf\right\vert}^2{\operatorname{id}_{}}$. Hence ${{A}^*}_\theta$ does not stem from a generalized complex structure provided that ${\left\vertf\right\vert}=1$.

Elliptic Lie algebroids
-----------------------

As in [@math/0509284], we say that a complex Lie algebroid ${B}$ is **elliptic** if $\operatorname{Re}\circ{{a}_{{B}}}:{B}\to{T{\boldsymbol{X}}}$ is surjective. Here ${{a}_{{B}}}:{B}\to{T_{{\mathbb{C}}}{\boldsymbol{X}}}$ is the anchor map of ${B}$ and $\operatorname{Re}:~{T_{{\mathbb{C}}}{\boldsymbol{X}}}\to{T{\boldsymbol{X}}}$ is the projection onto the real part.

\[extremely\] If ${B}$ is an elliptic Lie algebroid over a compact complex 