Proposition 1

Let V be a finite dimensional $\mathfrak{g}$-module.

(i) All the weights of V belong to P.

(ii) $V =\bigoplus_{\mu\in P}V^\mu$.

(iii) For all $\mu\in \mathfrak{h}^*,V^\mu$ is the set of $x\in V$ such that $h.x=\mu(h)x$ for all $h\in \mathfrak{h}$.

For all $\alpha \in B$, there exists a homomorphism from $\mathfrak{s}\mathfrak{l}(2, k)$ to $\mathfrak{g}$ that takes $H$ to $H_{\alpha}$. Thus, by §1, no. 2, Cor. of Prop. $2, (H_{\alpha})_V$ is diagonalizable and its eigenvalues are integers. Hence, the set of $(H_{\alpha})_V$, for $\alpha \in B$, is diagonalizable (Algebra, Chap. VII, §5, no. 6, Prop. 13). Consequently, for all $h\in \mathfrak{h},h_V$ is diagonalizable. By Chap. VII, §1, no. 3, Prop. $9, V =\bigoplus_{\mu\in\mathfrak{h}^*}V^\mu$. On the other hand, if $V^\mu\not= 0$, the preceding shows that $\mu(H_{\alpha})\in \mathbf{Z}$ for all $\alpha \in B$, so $\mu\in P$. This proves (i) and (ii). We see in the same way that $\mathfrak{h}_V$ is diagonalizable, hence (iii).