Is the adjoint representation group homomorphism injective?

Question

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?

Answer 1

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.

Answer 2

Quite obviously $Z(G) \subseteq ker(Ad)$. Think about when equality holds (hint: $G$ connected?).