Proposition 1

Let g be a Lie algebra over K and a an ideal of g. Let M be a g-module and $N$ a simple $a$-module. Consider $M$ as an $a$-module and suppose that the set of sub-$a$-modules of $M$ which are pure of species $N$ admits a maximal element $M'$. Then $M'$ is a sub-$g$-module of $M$.

Let $y \in g$. Let $\phi$ be the canonical mapping of $M$ onto $M/M'$ and $f$ the mapping $m \mapsto \phi(y_M \cdot m)$ of $M'$ into $M/M'$. It suffices to show that $f(M') = \{0\}$. Let $x \in a$. Then, for $m \in M$,

Now $[x, y] \in a$, whence $\phi([x, y]_M \cdot m) = 0$; on the other hand,

Hence $x_{M/M'} \cdot f(m) = f(x_M \cdot m)$. It follows that $f(M')$ is a sub-$a$-module of $M/M'$ isomorphic to a quotient of $M'$ and hence pure of species $N$; hence $f(M') = \{0\}$.