Discrete subgroups of solvable Lie groups of type $(E)$

Let $G_1$ and $G_2$ be simply connected Lie groups, and let $\Gamma$ be a lattice in $G_1$. In the present article we investigate the question whether the homomorphism $\mu\colon\Gamma\to G_2$ can be lifted to a homomorphism $\mu\colon G_1\to G_2$ for the case that $G_1$ or $G_2$ is a Lie group of type $(E)$. Incidentally we prove some of the properties of lattices in such groups.
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