Let $G$ be a semisimple real Lie group that is linear, i.e. admits a closed embedding $G \hookrightarrow \text{GL}(n,\mathbb{R})$ for some $n \in \mathbb{N}$. Does $G$ have only finitely many connected components?
If $G$ is real algebraic, then this is known to be true. On the other hand, if we drop the linearity condition, then $\text{SL}(2,\mathbb{R}) \times \mathbb{Z}$ is a counterexample.
Related to this questions is this question and the comments there.
This is false, just like the other question, because $\mathbb Z$ is a linear Lie group. It embeds in $\operatorname{GL}_1(\mathbb R)$ by $n \mapsto 2^n$.