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If $G$ is a Lie group with $Z(G)=\{g\in G:hg=gh \;\;\;\text{for all $h\in H$}\}=\{e\}$

Then is $G$ necessarily connected?

I tried to find some counterexamples to no avail; it seems quite hard to find centerless Lie groups in the first place.

On the other hand, I think $G$ might be path-connected through paths to $e$.

Any helps are appreciated!

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No: Take any centerless connected Lie group (e.g. $\mathrm{SO}(3)$), any centerless finite group (e.g. $S_3$) and take their cartesian product. So for example, $$\mathrm{SO}(3)\times S_3$$ has no centre but six connected components.

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    It might sound a bit stupid, but how do you know resulting cartesian product is a Lie group? – 2017-02-20
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    @user160738 A product of Lie groups is a Lie group and all finite groups are $0$-dimensional Lie groups. – 2017-02-20
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    @user160738 So the finite group $S_3$ alone is also an example, but I guess you wanted an example with positive dimension. – 2017-02-20
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    Got it, thanks! – 2017-02-20