Exercise 5, § 5

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. Then $\mathfrak{g}$ is said to be algebraic if, for all $x\in \mathfrak{g}$, the replicas of $x$ (Chap. I, §5, Exerc. 14) belong to $\mathfrak{g}$. Such an algebra is decomposable. a) Denote by $a(\mathfrak{g})$ the smallest algebraic subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ containing $\mathfrak{g}$. Then

Give an example where $a(\mathfrak{g})$ and $e(\mathfrak{g})$ are distinct (take V of dimension 2 and $\mathfrak{g}$ of dimension 1).

b) Show that, if $\mathfrak{n}$ is an ideal of $\mathfrak{g},\mathfrak{n}$ and $a(\mathfrak{n})$ are ideals of $a(\mathfrak{g})$, and $[a(\mathfrak{g}), a(\mathfrak{n})] = [\mathfrak{g},\mathfrak{n}]$ (imitate the proof of Prop. 4). Deduce that $\mathscr{D}^ia(\mathfrak{g}) =\mathscr{D}^i\mathfrak{g}$ for $i\geq 1$ and that $\mathscr{C}^ia(\mathfrak{g}) =\mathscr{C}^i\mathfrak{g}$ for $i\geq 2$.

c) Show that every Lie algebra consisting of nilpotent elements is algebraic.$^8$