Baer Lie property is not quotient-closed

This article gives the statement, and possibly proof, of a group property (i.e., Baer Lie group) not satisfying a group metaproperty (i.e., quotient-closed group property).
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Statement

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

Related facts

• Lazard Lie property is not quotient-closed

Proof

Further information: unitriangular matrix group:UT(3,Q), quotient of UT(3,Q) by a central Z

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 Baer Lie group: it is a group of nilpotency class exactly two that has 2-torsion.