Exercise 11, § 2

Let $Y$ and $Z$ be two disjoint sets (with $Y$ totally ordered), let $X = Y \cup Z$ and $l = L(X)$. Let $a_Y$ be the submodule of $l$ generated by $Z$ and $[l, l]$; it is an ideal of the Lie algebra $l$. We propose to find explicitly a basic family of $a_Y$.

Let $M$ be the set of positive integral functions on $Y$ of finite support. If $m \in M$, we denote by $\theta^m$ the endomorphism of $l$ given by

where the product is relative to the order relation given on $Y$. Let $\mathscr{S}$ denote the subset of $Y \times M$ consisting of the ordered pairs $(u, m)$ such that there exists $y \in Y$ for which $y > u$ and $m(y) \geq 1$. Show that the elements form a basic family of the Lie algebra $a_Y$.

(Reduce the problem to the case where $Y$ is finite and argue by induction on $\operatorname{Card}(Y)$. Use the Corollary to Proposition 10 applied to the greatest element $y$ of $Y$ and apply the induction hypothesis to $Y - \{y\}$.)