Corollary

Let $g$ be a Lie algebra over $K$ and $a$ an ideal of $g$. Let $M$ be a simple $g$-module, of finite length as a $K$-module. There exists a simple $a$-module $N$ such that $M$ is a pure $a$-module of species $N$.

Since the $a$-module $M$ is of finite length, there exists a minimal element $N$ in the set of sub-$a$-modules of $M$: it is a simple sub-$a$-module of $M$. The largest sub-$a$-module of $M$ which is pure of species $N$ is therefore $\neq \{0\}$ and is a sub-$g$-module of $M$ (Proposition 1) and is therefore identical with $M$.