Exercise 22, § 6

(a) Let $g$ be a nilpotent Lie algebra over an algebraically closed field K of arbitrary characteristic. Let $z$ be its centre, $\rho$ a finite-dimensional representation of $g$ and $\beta$ the associated bilinear form. Show that $z \cap \mathscr{D}g$ is orthogonal to $g$ with respect to $\beta$. (Reduce it, using a Jordan-Hölder series, to the case where $\rho$ is simple. Then note that, for $z \in z \cap \mathscr{D}g$, $\rho(z)$ is a scalar matrix with trace zero. Then use Exercise 22 (b) of § 4.) Deduce that, if $\beta$ is non-degenerate, $g$ is commutative. (Use Exercise 3 (a) of § 4.)

(b) Suppose that K is of characteristic 2. Show that for the solvable Lic algebra $\mathfrak{gl}(2, K) = \mathfrak{g}$ the bilinear form associated with the identity representation is non-degenerate, but that the centre $\mathfrak{z}$ is contained in $\mathscr{D}\mathfrak{g}$ and is $\neq \{0\}$.

(c) Let $\mathfrak{g}$ be the 6-dimensional Lie algebra over $K$ with basis $(a, b, c, d, e, f)$ and multiplication table $[a, b] = -[b, a] = d,\ [a, c] = -[c, a] = e,\ [b, c] = -[c, b] = f$ and the other brackets are 0. Let $\beta$ be the bilinear form on $\mathfrak{g}$ such that $\beta(c, d) = \beta(d, c) = 1,\ \beta(a, f) = \beta(f, a) = 1,\ \beta(b, e) = \beta(e, b) = -1$ and the other values of $\beta$ at ordered pairs of the basis of $\mathfrak{g}$ in question are 0. Show that $\beta$ is invariant, $\mathfrak{g}$ nilpotent, $\mathfrak{z} = \mathscr{D}\mathfrak{g} \neq \{0\}$ and $\beta$ non-degenerate.