Exercise 23, § 9

(a) Let G be a finite-dimensional commutative real Lie group with only a finite number of connected components. For G to be compact, it is necessary and sufficient that every finite-dimensional analytic linear representation of G on a complex vector space be semi-simple. (Use the proof of Proposition 32.)

(b) Let G be a finite-dimensional commutative complex Lie group with only a finite number of connected components. Let N be the kernel of $\exp_G$.

For N to generate over $\mathbf{C}$ the vector space $L(G)$, it is necessary and sufficient that every finite-dimensional analytic linear representation of G be semi-simple. (Use (a), Lemma 1 and Proposition 32.)