Proposition 9

Let $E,F$ be finite dimensional simple $\mathfrak{g}$-modules, with highest weights $\lambda , \mu$, respectively.

(i) The component of $E\otimes F$ of highest weight $\lambda +\mu$ is a simple submodule, generated by $(E\otimes F)^{\lambda+\mu}= E^{\lambda}\otimes F^\mu$.

(ii) The highest weight of any simple submodule of $E\otimes F$ is $\leq \lambda +\mu$ (cf. §9, Prop. 2).

If $\alpha , \beta \in P$ and if $E^{\alpha}\otimes F^{\beta}\not= 0$, then $\alpha \leq \lambda$ and $\beta \leq \mu$. Consequently, $(E\otimes F)^{\lambda+\mu}$ is equal to $E^{\lambda}\otimes F^\mu$, and hence is of dimension 1, and $\lambda +\mu$ is the highest weight of $E\otimes F$. Every non-zero element of $E^{\lambda}\otimes F^\mu$ is primitive. By Prop. 4 of §6, no. 2, the length of the isotypical component of $E\otimes F$ of highest weight $\lambda +\mu$ is 1.