Baer Lie property is not subgroup-closed

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This article gives the statement, and possibly proof, of a group property (i.e., Baer Lie group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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It is possible to have a Baer Lie group (a 2-powered class two group) G and a subgroup H of G that is not a Baer Lie group.

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Further information: unitriangular matrix group:UT(3,Q), unitriangular matrix group:UT(3,Z)

Consider the example G = UT(3,\mathbb{Q}) and H = UT(3,\mathbb{Z}). G is a Baer Lie group. H is a group of class exactly two that is not 2-powered, hence, it is not a Baer Lie group.