Exercise 16, § 2

Let $\mathfrak{g}$ be a solvable complex Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$, $\mathfrak{g}=\bigoplus\mathfrak{g}^{\lambda}(\mathfrak{h})$ the corresponding decomposition of $\mathfrak{g}$ into primary subspaces, with $\mathfrak{g}^0(\mathfrak{h}) =\mathfrak{h}$.

a) Show that the restrictions to $\mathfrak{h}$ of the linear forms called roots of $\mathfrak{g}$ in Chap. III, §9, Exerc. $17c)$ are the weights of $\mathfrak{h}$ in $\mathfrak{g}$, i.e. the $\lambda$ such that $\mathfrak{g}^{\lambda}(\mathfrak{h})\not= 0$; deduce that such a $\lambda$ vanishes on $\mathfrak{h}\cap \mathscr{D}\mathfrak{g}$.

b) Let $(x, y) \rightarrow [x, y]'$ be the alternating bilinear map from $\mathfrak{g}\times \mathfrak{g}$ to $\mathfrak{g}$ with the following properties:

(i) If $x\in \mathfrak{g}^{\lambda}(\mathfrak{h}), y\in \mathfrak{g}^\mu(\mathfrak{h})$, with $\lambda \not= 0, \mu\not= 0$, then $[x, y]'= [x, y]$;

(ii) if $x\in \mathfrak{g}^0(\mathfrak{h}), y\in \mathfrak{g}^\mu(\mathfrak{h})$, then $[x, y]'= [x, y]-\mu(x)y$.

Show that this gives a new Lie algebra structure on $\mathfrak{g}$ (use $a$)$)$. Denote it by $\mathfrak{g}'$.

c) Show that, if $x\in \mathfrak{g}^{\lambda}(\mathfrak{h})$, the map ad$'x:y \rightarrow [x, y]'$ is nilpotent. Deduce that $\mathfrak{g}'$ is nilpotent (apply Exerc. 11 of Chap. I, §4 to the set E of ad$'x$, where $x$ belongs to the union of the $\mathfrak{g}^{\lambda}(\mathfrak{h})$).