  We obtain all extreme and exposed points of the closed unit ball of the space
of 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 optimal
constant of the Littlewood's $4/3$ inequality is achieved is necessarily an
extreme point. In the case of complex scalars we combine analytical and
numerical evidences supporting that, at least for complex bilinear forms with
real coefficients, the optimal constant of Littlewood's $4/3$ inequality seems
to the the trivial, i.e., $1.$
