Talk:Lie groups I: Definition and examples
From Manifold Atlas
Theorem 2.2 is wrong as stated. A counterexample is given on page 83 of [Carter&Segal&Macdonald1995].
For each Lie group one has the adjoint representation $Ad:G\rightarrow GL\left(g\right)$. This representation is faithful if $G$ is semisimple. Thus Theorem 2.2 is correct for semisimple Lie groups.
I'm going to add the condition "G semisimple" to the assumptions of Theorem 2.2.