Today I am grateful because I learned that "a semisimple Lie subgroup of a simply connected Lie group is always closed [1] pg 615" and I needed it for my research. :D

[1]: https://www.jstor.org/stable/pdf/1969437.pdf?casa_token=zt2jzD1KEMAAAAAA:p-MqIoXoCEzk4bpr_bAB4Hfl3vfqde5oc-cdubmFFZk6ySjejd2a77c6fg66B5LfXfLefKse6bX_lv5IitFBzRTkWCKAV3C-ZDJoQDbHCF_49bWI0Hf7Sg