Proposition 4

Let V be a simple $\mathfrak{g}$-module with a highest weight $\omega$, X a semi-simple $\mathfrak{g}$-module, and $X'$ the isotypical component of type V in X. Then $X'$ is the submodule of X generated by $X^{\omega}_{\pi}$. Its length is equal to the dimension of $X^{\omega}_{\pi}$.

Let $X''$ be the submodule of X generated by $X^{\omega}_{\pi}$. It is clear that every submodule of X isomorphic to V is contained in $X''$. Hence $X'\subset X''$. On the other hand, let $\Phi$ be the set of homomorphisms from the $\mathfrak{g}$-module V to the $\mathfrak{g}$-module X. The length of $X'$ is dim$_k\Phi ($Algebra, Chap. VIII, §4, no. 4), that is dim$_kX^{\omega}_{\pi}$ (Prop. 2).