Exercise 11, § 7

Let K be a commutative field in which $2 = 0$, A the polynomial ring $K[X, Y, Z]$ and E the A-module $A^3/M$, where M is the submodule of $A^3$ generated by the element $(X, Y, Z)$ of $A^3$; finally, let F be the quotient A-module of A by the ideal $AX^2 + AY^2 + AZ^2$. Show that there exists an alternating bilinear mapping of $E^2$ into F, which is the skewsymmetrization of no bilinear mapping of $E^2$ into F, but that in $T^2(E)$ the module $\mathfrak{g}_2''$ is equal to the kernel of the endomorphism $z \mapsto az$ (consider $T^2(E)$ as a quotient module of $T^2(A^3)$ and show that the kernel of $z \mapsto az$ is the canonical image of the module of symmetric tensors $S'_2(A^3)$).