  Inspired by an article of R. Bryant on holomorphic immersions of unit disks
into Lorentzian CR manifolds, we discuss the application of Cartan's method to
the 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 that
the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie
in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then
provide an example where one of the functions does not identically vanish, thus
obstructing holomorphic immersions.
