TITLE: Is there a non-matrix Lie group?
QUESTION [18 upvotes]: I'm new to Lie Groups, but all the examples I found are matrix groups. Can someone show a non-matrix Lie group?

REPLY [8 votes]: There are no non-matrix Lie groups whose dimension is $1$ or $2$. On the other hand, consider the quotient of the Heisenberg group$$\left\{\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}\,\middle|\,a,b,c\in\mathbb R\right\}$$by the normal subgroup$$\left\{\begin{pmatrix}1&0&m\\0&1&0\\0&0&1\end{pmatrix}\,\middle|\,m\in\mathbb Z\right\}.$$It is a non-matrix three-dimensional Lie group.