Lie algebra od $Ad(G)$

Question

Let $G$ be a Lie group. Let $\mathfrak g$ be the Lie algebra of $G.$ We have $Ad:G\to Aut(\mathfrak g)$ adjoint representation of $G.$ Let $K$ be a compact connected Lie subgroup of $G$ with Lie algebra $\mathfrak k$. I want to show that the Lie algebra of $Ad(K)$ which is itself a Lie group has the Lie algebra $ad(\mathfrak k).$ For any $ad X$ with $X\in\mathfrak k$ we have that $exp(tX)\in K\subseteq G.$ Then as $ad$ is just the differential of $Ad$ we can show that $adX$ is in the Lie algebra of $Ad(K).$ How to prove the converse inclusion?