Eqs. - mean that $\pi$ is a holomorphic Poisson tensor with respect to the deformed complex structure.

\[Cor:Hminus\] If $H=\pi+\theta+\omega$ is an extended Poisson structure, then so is $$\lambda\pi+\theta+\lambda{^{-1}}\omega ,$$ for any $\lambda\in{\mathbb{C}}^\times$. In particular, $$H^{\vee}=-\pi+\theta-\omega$$ is an extended Poisson structure.

Note that Maurer-Cartan elements as deformations of Lie bialgebroids or differential Gerstenhaber algebras were already considered by Cleyton-Poon [@CP] in their study of nilpotent complex structures on real six-dimensional nilpotent algebras.

A natural question is: when will $({A},{{A}^*}_H)$ arise from a generalized complex structure in the sense of Hitchin [@MR2013140; @GualtieriThesis]? Let us recall the following:

(Lemma 6.1 in [@arxivmath0702718]) The graph $\{H{^{\sharp}}\xi+\xi\in{A}\oplus {{A}^*}\}$ of $H$, which is clearly isomorphic to ${{A}^*}_H$ as a vector bundle, is the $+{i}$- (or $-{i}$-) eigenbundle of a generalized complex structure on ${\boldsymbol{X}}$ if and only if ${\overline{H}{^{\sharp}}}\circ {H{^{\sharp}}}-{\operatorname{id}_{{{A}^*}}}$ is invertible. Here the map ${\overline{H}{^{\sharp}}}:{A}\to{{A}^*}$ is defined by ${\overline{H}{^{\sharp}}}(u)=\overline{{H{^{\sharp}}}(\overline{u})}$, $\forall u\in{A}$.

Again we let $H=\pi+\theta+\omega$ be an extended Poisson structure on ${\boldsymbol{X}}$. Relatively to the direct sum decompositions \[eq3\] of ${A}$ and ${{A}^*}$, the endomorphisms ${H{^{\sharp}}}$ and ${\overline{H}{^{\sharp}}}$ are represented by the block matrices $${H{^{\sharp}}}= \left( \begin{matrix} \theta{^{\flat}}& \pi{^{\sharp}}\\ 
\omega{^{\flat}}& \theta{^{\sharp}}\end{matrix} \right) 
\quad \text{and} \quad 
{\overline{H}{^{\sharp}}}= \left( \begin{matrix} {\overline{\theta}{^{\flat}}}& {\overline{\pi}{^{\sharp}}}\\ 
{\overline{\omega}{^{\flat}}}& {\overline{\theta}{^{\sharp}}}\end{matrix} \right) 