A connected semisimple Lie subgroup of $SO(n)$ is closed in $SO(n)$ (Kobayashi and Nomizu, 1963, p. 279). Can we extend this result to all semisimple groups, put differently, is any connected semisimple Lie subgroup of a semisimple group closed in this semisimple group?
I thought that maybe I could use arguments like "any connected semisimple group is faithfully represented as subgroup of $SO(n)$" which would immediately prove the extension to all semisimple group. A push in the right direction or reference would be enough.
Yes, it is true. It is a consequence of the facts below.
If $h$ is a Lie subalgebra of $gl(n,K)$ ($K = \mathbb{R}$ or $\mathbb{C}$) then $[h,h]$ is the subalgebra of an algebraic subgroup of $GL(n,K)$ hence corresponds to a (connected) closed subgroup ( an old result of Chevalley and Tuan.
If $G$ has a morphism $\phi$ to a $GL(n,K)(=G_1) $ with an injective tangent map $d\phi$ and $d\phi(h)$ corresponds to a closed subgroup $H_1$ in $G_1$ then $h$ corresponds to a closed subgroup $\phi^{-1}(H_1)$.
From 1., every semisimple Lie subalgebra of a $GL(n,K)$ corresponds to an algebraic subgroup hence a closed subgroup.
For 2., map consider $\phi \colon G \to GL(g)$, $g \mapsto Ad(g)$, with differential $d\phi=ad$ injective, since $g$ has zero center.