Let $M$ be a smooth manifold and let $TM\otimes\mathbb C$ be its complexified tangent bundle. Let $H$ be a complex subbundle of $TM\otimes\mathbb C$.
It can be happened that $H$ and $\bar H$ are involutive (i.e $[H,H]\subset H$ and $[\bar H,\bar H]\subset \bar H$) but $H\oplus \bar H$ is not involutive.
Can you please provide me in examples of this situation?