It is known that a discrete normal subgroup $N$ of the connected group $G$ is contained in the center of $G$. But we also know that if a group is discrete then its Lie algebra is $\{0\}$, and we know that if $G$ is connected then $\exp(\mathfrak{g})$ generates $G$, where $\mathfrak{g}$ is its Lie algebra.
So consider the Lie algebra of the subgroup $N$: this has to be $\{0\}$, and so $N$ has to be $\{1\}$, which means $N$ must be in the center. However this seems false somehow. Where exactly am I going wrong?