Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if 


1. $P$ is Lagrangian
2. P involutive
3. dim$P\cap\bar P \cap TM$ is constant.


Let $P\_1, P\_2$ be two complex polarization, then when $\bar P\_1\cap P\_2$ is involutiove ?



A question about complex polarization
I may be missing something, but isn't it completely obvious that $\bar P\_1\cap P\_2$ is always involutive?


Indeed, if $\xi$, $\eta$ are two complex vector fields taking values in $\bar P\_1\cap P\_2$, then $[\xi,\eta]$ takes its values in $\bar P\_1$ because $\bar P\_1$ is involutive, and in $P\_2$ because $P\_2$ is involutive, hence it takes values in $\bar P\_1\cap P\_2$, no?


