Exercise 13, § 2

(In this exercise we assume that every projective module over $K$ is free; this is so for example if $K$ is a principal ideal domain.)

Let $l = \sum_{n=1}^\infty l_n$ be a graded Lie algebra admitting a basic family $B$ consisting of homogeneous elements and let $b$ be a graded Lie subalgebra of $l$, which is a direct factor of $l$ considered as a module.

(a) For all $i \geq 0$, let $l^{(i)}$ be the graded subalgebra of $l$ such that

Then $l = l^{(0)} \supset l^{(1)} \supset \cdots \supset b$. Show, arguing by induction on $i$, the existence of a basic family $B^{(i)}$ of $l^{(i)}$, consisting of homogeneous elements, such that the elements of $B^{(i-1)}$ and of $B^{(i)}$ of degree $< i - 1$ are the same. (Suppose that $B^{(i-1)}$ has been constructed. Let $m_i$ be the intersection of $l_i = l_i^{(i-1)}$ with the subalgebra generated by the $b_j$ for $j < i$ and let $b_i$ be the submodule of $l_i$ generated by the elements of $B^{(i-1)}$ of degree $i$. The induction hypothesis implies that $l_i = m_i \oplus b_i$. As $m_i \subset b_i$, we can decompose $b_i$ as $b_i = \eta_i \oplus \delta_i$, so that $b_i = m_i \oplus \delta_i$; by the hypothesis on $K$ the modules $\eta_i$ and $\delta_i$ are free;

changing the elements of $B^{(i-1)}$ of degree $i$ if necessary, we can assume that $\mathfrak{z}_i$ is generated by a subset of $B^{(i-1)}$. Applying Exercise 11 to $\mathfrak{l}^{(i-1)}$ and its ideal $\mathfrak{l}^{(i)}$, we obtain a basic family $B^{(i)}$ of $\mathfrak{l}^{(i)}$ with the desired properties.)

(b) Deduce from (a) the fact that $\mathfrak{h}$ has a basic family consisting of homogeneous elements.