Exercise 5, § 7

Let E be a $\mathfrak{g}$-module. Show the equivalence of the conditions:

(i) E is faithful (i.e. the canonical map from $\mathfrak{g}$ to $\mathfrak{g}\mathfrak{l}(E)$ is injective).

(ii) Every root of $\mathfrak{g}$ is the difference between two weights of E.