Corollary 4

Let G be a finite-dimensional connected Lie group. The following conditions are equivalent: (i) G is unimodular (Integration, Chapter VII, § 1, no. 3, Definition 3); (ii) det Ad g = 1 for all g \in G; (iii) Tr ad a = 0 for all a \in L(G).

The mapping g \mapsto det Ad g is a morphism $\phi$ of G into K*. By § 3, Propositions 35 (no. 10) and 44 (no. 12), $L(\phi)a = Tr\ ad\ a$ for all $a \in L(G)$. Clearly Im $L(\phi) = \{0\}$ or K. In the first (resp. second) case, Im $\phi = \{1\}$ (resp. Im $\phi = K^*$) by Corollary 1 and hence G is unimodular (resp. not unimodular) by § 3, no. 16, Corollary to Proposition 55.