Exercise 14, § 1

Let $a$ be a Lie algebra and $b$ a subalgebra of $a$. A composition series joining $a$ to $b$ is a decreasing sequence $(a_i)_{0 \leqslant i \leqslant n}$ of subalgebras of $a$ such that $a_0 = a, a_n = b, a_{i+1}$ is an ideal of $a_i$ for $0 \leqslant i < n$. $b$ is called a subinvariant subalgebra of $a$ if there exists a composition series joining $a$ to $b$. (a) Let $b$ be a subinvariant subalgebra of $a$ and $(a_i)_{0 \leqslant i \leqslant n}$ a composition series joining $a$ to $b$. Deduce from Exercise 13 that $[\mathscr{C}^{n+p}b, a] \subset \mathscr{C}^p b$ by observing that $[a_i, b] \subset a_{i+1}$ for $1 \leqslant i < n$. (b) Deduce from (a) that the intersection $\mathscr{C}^\infty b$ of the $\mathscr{C}^p b$ ($p = 0, 1, 2, \ldots$) is an ideal of $a$. (c) Deduce from (b) that a subinvariant subalgebra $c$ of $a$ such that $c = \mathscr{D}c$ is an ideal of $a$. (d) When $K$ is a field and $\dim_K a < +\infty$, show that the intersection $\mathscr{D}^\infty b$ of the $\mathscr{D}^p b$ ($p = 0, 1, 2, \ldots$) is an ideal of $a$. (Show that this intersection is a subinvariant subalgebra of $a$ and apply (c).)