Exercise 9, § 2

Let k be the field with two elements, V the vector space $k^3$, $(x_1, x_2, x_3)$ its canonical basis and K the exterior algebra of V, which is an 8-dimensional commutative algebra over k. Let g be the Lie K-algebra admitting the basis $(e_1, e_2, e_3, e_{12}, e_{13}, e_{23})$ such that

and the other brackets are zero. Let $\mathfrak{h}$ be the ideal of g generated by $u = x_1 e_1 + x_2 e_2 + x_3 e_3$. As a module, $\mathfrak{h}$ is generated by u, $[u, e_1] = x_2 e_{12} + x_3 e_{13}$, $[u, e_2] = x_1 e_{12} + x_3 e_{23}$, $[u, e_3] = x_1 e_{13} + x_2 e_{23}$. Let (a) Show that $v \notin \mathfrak{h}$. (Consider the K-linear form $\phi$ on g such that $\phi(e_1) = \phi(e_2) = \phi(e_3) = 0, \phi(e_{12}) = x_3, \phi(e_{13}) = x_2, \phi(e_{23}) = x_1.$) (b) Let f be a linear mapping of g into an associative K-algebra such that $f([x, y]) = f(x) f(y) - f(y) f(x)$ for all x, y in g. Show that $f(v) = f(u)^2$. (c) Deduce from (a) and (b) that the canonical mapping of the Lie algebra $g/\mathfrak{h}$ into its enveloping algebra is not injective.