  Let $X$ be a Banach space and let $\kappa(X)$ denote the kernel of a quotient
map $\ell_1(\Gamma)\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if
bilinear forms on $\kappa(X)$ extend to $\ell_1(\Gamma)$. From that we obtain
i) If $\kappa(X)$ is a $\mathcal L_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$
is separable, $\kappa(X)$ is not an $\mathcal L_1$ space and $\Ext^2(X,X^*)=0$
then $\kappa(X)$ has an unconditional basis. This provides new insight into a
question of Palamodov in the category of Banach spaces.
