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We obtain all extreme and exposed points of the closed unit ball of the spaceof bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow\mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow\mathbb{R}$ for which the optimalconstant of the Littlewood's $4/3$ inequality is achieved is necessarily anextreme point. In the case of complex scalars we combine analytical andnumerical evidences supporting that, at least for complex bilinear forms withreal coefficients, the optimal constant of Littlewood's $4/3$ inequality seemsto the the trivial, i.e., $1.$

Geometry of the closed unit ball of the space of bilinear forms on $\ell_{\infty}^{2}$