Exercise 3, § 7

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

(i) The rank of the semi-direct product of $\mathfrak{g}$ by E is strictly larger than that of $\mathfrak{g}$.

(ii) 0 is a weight of E.

(iii) There exists a weight of E that is radical (i.e. belongs to Q).

b) Assume that E is simple. Show that (i), (ii), (iii) are equivalent to

(iv) The highest weight of E is radical.

If these conditions are satisfied, there exists no non-zero invariant alternating bilinear form. (Use Prop. 12 and Prop. 1 (ii) of §6.)