Image of the adjoint representation of a Lie algebra

Question

Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$ and $K$ a Lie subgroup with Lie algebra $\mathfrak k \subset \mathfrak g$.

Can we prove that $\text{Ad}_G(K)$ is generated by $\text{Ad}_G(\text{exp(X)})$ for $X \in \mathfrak k$?

Answer 1

No, this is not true. It would say that a discrete subgroup $K\subset G$ of a connected Lie group would automatically lie in the kernel of $Ad_G$ and hence in the center of $G$. But this is not true at all, just take $SL(2,\mathbb Z)\subset SL(2,\mathbb R)$ as an example.