Proposition 3

Let G be a Lie group of which the group of connected components is finite. The following conditions are equivalent:

(i) The Lie algebra L(G) is compact.

(ii) The group Ad(G) is compact.

(iii) There exists a separating positive symmetric bilinear form on L(G) invariant under the adjoint representation of G.

\*(iv) G has a riemannian metric invariant under left and right translations.\*

(i) $=\Rightarrow$ (ii): if L(G) is compact, the group Ad(G$_0$) $=$ Int(L(G)) is compact; since it has finite index in Ad(G), this latter group is also compact.

(ii) $=\Rightarrow$ (iii): this follows from no. 1.

(iii) $=\Rightarrow$ (i): since Int(L(G)) $\subset$ Ad(G), this follows from the characterization (iii) of Prop. 1.

\*(iii) $\Leftarrow \Rightarrow$ (iv): this follows from Chap. III, §3, no. 13.\*