Let $G$ be a Lie group and let $H$ be a closed subgroup such that $g_1g_2\in H$ for all $g_1,g_2\in H^c$. Now since $H^c$ is open then it's a manifold and actually we have transitive action of $H$ on $H^c$ by the group multiplication. The isotropy group is $\{e\}$. Thus, $H$ is diffeomorphic to $H^c$.
An example of this is the orthogonal group $O(n)$, since the special orthogonal group $SO(n)$ is a closed subgroup and satisfies the property above then $SO(n)\cong SO(n)^c$.
My question: could this be interesting somewhere in geometry? In general what are those Lie groups have such normal subgroups? And what Tits fibration would be?
I think you can already see from the facts you observed that this is a very restrictive situation. You basically have observed that multiplication by an element of $H^c$ is a diffeomorphism $G\to G$ which maps $H$ diffeomorphically onto $H^c$ and $H^c$ diffeomorphically onto $H$. So $H$ is both open and closed. Hence it is a union of some connected components of $G$ and in particular has to contain the connected component $G_0$ of the identity. Moreover, $G/H=\mathbb Z_2$ so the situation is very close to the one of $O(n)$ you describe.