While doing some exercises in Lie groups, I see that the Lorentz group $O(1,3)$ has four connected components and $\pi_0(O(1,3))$ is the Klein four-group $\mathbb{Z}/2 \times \mathbb{Z}/2$. Not only that, but I can find explicit elements representing each connected component $\{1,a,b,ab\}$ which form a subgroup isomorphic to $\pi_0$.

My question then: is it true that for any Lie group $G$, the group $\pi_0(G)$ embeds as a subgroup of $G$? I believe it would follow that any such embedding would take distinct elements of $\pi_0$ to distinct components of $G$.