Lazard Lie property is not quotient-closed

Statement
It is possible to have a Lazard Lie group $$G$$ and $$H$$ a normal subgroup of $$G$$ such that the quotient group $$G/H$$ is not a Lazard Lie group.

Proof
Suppose $$G$$ is the group $$UT(3,\mathbb{Q})$$, the unitriangular matrix group of degree three over the field of rational numbers. Let $$H$$ be a central subgroup of $$G$$ that is isomorphic to $$\mathbb{Z}$$, the group of integers. The quotient group $$G/H = UT(3,\mathbb{Q})/\mathbb{Z}$$ is not a Lazard Lie group: it is a group of nilpotency class exactly two that has 2-torsion.