# Talk:Lie groups I: Definition and examples

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.

Nachtrag: Now that the statement has changed to include compact Lie groups only: wouldn't it be clearer to state that compact Lie groups are isomorphic to subgroups of O(n) not just of GL(n,R)?

Or perhaps one should divide the theorem into two parts: a) each semisimple Lie group is isomorphic to a subgroup of GL(n,R) b) each compact. Lie group is isomorphic to a subgroup of O(n)