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Inspired by an article of R. Bryant on holomorphic immersions of unit disksinto Lorentzian CR manifolds, we discuss the application of Cartan's method tothe question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth$9$-dimensional real analytic real hypersurface $M^{9}\subset\mathbb{C}^{5}$with Levi signature $(2,2)$ passing through a fixed point. The result is thatthe lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must liein the zero set of two complex-valued functions in $M^{9}\times U(2)$. We thenprovide an example where one of the functions does not identically vanish, thusobstructing holomorphic immersions.

Holomorphic immersions of bi-disks into $9$ dimensional real hypersurfaces with Levi signature $(2, 2)$