Exercise 19, § 4

(a) Let g be a Lie algebra. Let $\mathfrak{D}$ be the Lie algebra of derivations of g. Characteristic ideals $g^{[k]}$ of g are defined inductively as follows: $g^{[0]} = g$ and $g^{[k+1]}$ is the subspace of g generated by the Dx ($D \in \mathfrak{D}, x \in g^{[k]}$). The following conditions are equivalent: (1) every derivation of g is nilpotent; (2) $g^{[k]} = \{0\}$ for k sufficiently large; (3) the holomorph of g is nilpotent. If they are fulfilled, g is said to be characteristically nilpotent. Such an algebra is nilpotent.

(b) Verify that an 8-dimensional Lie algebra is defined by the following multiplication table:

and $[x_i, x_j] = 0$ for $i + j > 8$. Show that

and that the transporter of $\mathscr{C}^2g$ into $\mathscr{C}^4g$ is $\sum_{i=2}^8 Kx_i$. Deduce that every derivation D of g is defined by formulae $Dx_i = \sum_{j=1}^8 u_{ij}x_j$. Then show that $u_{ii} = 0$ for all i if the characteristic of K differs from 2, so that g is characteristically nilpotent.

(c) If the characteristic of K is $\neq 2$, show that the Lie algebra g of (b) is not the derived Lie algebra of any Lie algebra. (If $g = \mathscr{D}\mathfrak{h}$, show first that $\mathfrak{h}$ is nilpotent. Observe that $\dim g/\mathscr{D}g = 2$, which with Exercise 7 (c) leads to a contradiction.)