example of a connected matrix lie group with a nondiscrete normal subgroup H? What is an example of a connected matrix lie group with a non discrete normal subgroup H such that its tangent space at the identity is the zero matrix?

Take the group $G$ to be $SO(2)\simeq S^1$, a $1$-dimensional circle. Since this group is abelian, any subgroup will be normal. Then take dense subgroup $H$ isomorphic to $\mathbb{Z}$, which is generated by the image of $\sqrt{2}$ (or any irrational number) in $S^1$ via the parametrization $\mathbb{R}\to S^1$, $x\mapsto e^{2\pi i x}$. Tangent space to $H$ will be zero since there are no non-constant curves $\gamma\colon \mathbb{R}\to G$ with image in $H$.
I think if you require $H$ to be closed, there can't be such pathological example.
