Consider a Lie group $G$ and the homomorphism $\mathrm{Ad}: G\to \mathrm {GL}(\mathfrak g)$ which maps to each element $g\in G$ its adjoint representation $\mathrm {Ad}_g$.
Is this homomorphism injective? If not, what are examples where it fails to be injective?
Best Answer
If $G$ is commutative, $Ad_g$ is the identity for every $g$ this implies that if $G=\mathbb{R}^n, n>1$ for example the adjoint representation is not injective.