Exercise 4, § 7

Let $g$ be a Lie algebra, $a$ a subinvariant subalgebra of $g$, $r(a)$ the radical of $a$ and $\rho$ a finite-dimensional representation of $a$. For $\rho$ to admit a finite-dimensional extension to $g$, it is necessary and sufficient that $\mathscr{D}g \cap r(a)$ be contained in the largest nilpotency ideal of $\rho$. (For the sufficiency proceed as follows; let $g = g_0 \supset g_1 \supset \cdots \supset g_k = a$ be a decomposition series of $g$ with the properties of Exercise 11 (b) of § 6. Show by induction the existence of an extension $\rho_i$ of $\rho$ to $g_i$, whose largest nilpotency ideal contains $\mathscr{D}g \cap r(g_i)$. In the notation of Exercise 11 (b) of § 6, the passage from $\rho_{i+1}$ to $\rho_i$ follows from Theorem 1 when $h_i$ is simple, or when $h_i$ is 1-dimensional and

When $\mathscr{D}g \cap r(g_{i+1}) \neq \mathscr{D}g \cap r(g_i)$, it can be assumed that $h_i \subset \mathscr{D}g \cap r(g_i)$. Then $r(g_i) = r(g) \cap g_i$ ($§ 5$, Exercise 10) and Theorem 1 can be applied.)