Subgroups of semisimple Lie groups

Question

Let $G$ be a connected semisimple Lie group.

1. If $N$ is a normal closed semisimple Lie subgroup of $G$ and $\Lambda$ is a closed discrete subgroup of $G$. Is $N\Lambda$ a closed subgroup of $G$?
2. If $H$ is a closed connected semisimple Lie subgroup of $G$. Is the normalizer of $H$ in $G$ semisimple?

Answer 1

No to both.

1: try with a choice of direct product $G=N\times H$...

2: take $G=\mathrm{SL}_4$ and take $H=\mathrm{SL}_2\times\mathrm{SL}_2$ embedded as group of block-diagonal matrices each of determinant 1. What is the normalizer then?