Exercise 10, § 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{r}$ its radical, $\mathfrak{n}$ its largest nilpotent ideal and $\mathfrak{a}$ one of its Levi subalgebras.

a) Put $\mathfrak{g}'=\mathfrak{n}+\mathscr{D}\mathfrak{g}$. Show that $\mathfrak{g}'=\mathfrak{n}\oplus \mathfrak{a}$. (Use the fact that $[\mathfrak{g},\mathfrak{r}]$ is contained in $\mathfrak{n}.$) If $\mathfrak{g}\not= 0$, then $\mathfrak{g}'\not= 0$.

b) Assume that $k$ is algebraically closed. Let $(V_i)_{i\in I}$ be the quotients of a Jordan-Hölder sequence of the $\mathfrak{g}$-module $\mathfrak{g}$ (with the adjoint representation). If $x\in \mathfrak{r}$, show that $x_{V_i}$ is a homothety and that $x_{V_i}= 0$ for all $i$ if and only if $x$ belongs to $\mathfrak{n}$. Deduce that an element $y\in \mathfrak{g}$ belongs to $\mathfrak{g}'$ if and only if Tr($y_{V_i}$) $= 0$ for all $i\in I$.

c) Denote by N the vector subspace of $\mathfrak{g}$ generated by the elements $x$ such that ad $x$ is nilpotent. Show that N is a subalgebra of $\mathfrak{g}$ (use the fact that N is stable under Aut$_e(\mathfrak{g})$). Show, by using b), that $N\subset \mathfrak{g}'$.

d) Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Assume that there exists a subset R of $\mathfrak{h}^*$ such that

an assumption which is satisfied, in particular, if $k$ is algebraically closed. Show that N then contains the $\mathfrak{g}^{\alpha}(\mathfrak{h}),\mathscr{D}\mathfrak{h}$ and $\mathfrak{n}$; deduce that N contains $\mathfrak{g}'$, so $N =\mathfrak{g}'$.

e) If $k$ is algebraically closed and $\mathfrak{g}\not= 0,\mathfrak{g}$ contains an element $x\not= 0$ such that ad $x$ is nilpotent. (Indeed, we then have $\mathfrak{g}'\not= 0.$)