Nilpotent Lie Group that is not simply connect nor product of Lie Groups? I have been trying to find for days a non-abelian nilpotent Lie Group that is not simply connected nor product of Lie Groups, but haven't been able to succeed.
Is there an example of this, or hints to this group, or is it fundamentally impossible? 
Cheers and thanks.

$SL(2;\mathbb{R})$ is nonabelian, has infinite cyclic fundamental group, and is not a product since (I believe) $\mathfrak{sl}(2;\mathbb{R})$ does not decompose as a direct sum of Lie algebras.
Edit: Wait, do you want just a nonabelian, non-product Lie group with nontrivial $\pi_1$, or do you want it nilpotent as well?
