Lie group not related to matrix group

Question

Does there exist a Lie group that doesn’t come from a matrix group? Of course covers or quotients of matrix groups might not be matrix groups, and you can always take products of things like these to get more non-matrix groups. But is there a non-matrix Lie group that cannot be obtained from matrix groups using these operations?

Answer 1

Ado theorem says that every Lie algebra ${\cal G}$ can be embedded to a matrix Lie algeba. This induces a covering map between the simply connected group whose Lie algebra is ${\cal G}$ and a matrix group therefore the quotient of the universal cover of a connected Lie group is a matrix group.