Example of a semisimple Lie group which is not an algebraic group? I wonder if there are any examples of semisimple Lie groups that couldn't correspond to any algebraic group, or actually every semisimple Lie group is algebraic?

In plainer terms:
Let $p\colon \tilde G\to G=SL(2,\mathbb{R})$ be a  cover map. Consider $\phi\colon \tilde G \to H$, a morphism of Lie groups, where $H$ is a complex Lie group.  The map $d \phi \colon  \tilde{\frak{g}} \to \frak{h}$ is a morphism of real Lie algebras. Extend it  a morphism of complex Lie algebras $\tilde{\frak{g}}_{\mathbb{C}}=sl(2,\mathbb{C})\to \frak{h}$. Now, the group $SL(2,\mathbb{C})$ is simply connected, so there exists $\psi\colon SL(2,\mathbb{C})\to H$ with the given map between Lie algebras. We conclude $\phi = \psi \circ p$,  that is, the morphism $\phi$ factors through $G= SL(2,\mathbb{R})$. As a conclusion, there does not exists an injective morphism of Lie groups $\tilde G\to GL(n, \mathbb{C})$ (or $GL(n, \mathbb{R})$).
So: $G\subset G_{\mathbb{C}}$ simply connected, $\tilde G\to G$ cover, then any map from $\tilde G$ to a complex Lie group factors through $G$.
