Lazard Lie property is not finite direct product-closed

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

It is possible to have two groups G_1 and G_2 that are both Lazard Lie groups, but such that the external direct product G_1 \times G_2 is not a Lazard Lie group.

Proof

Take G_1 to be a Baer Lie group that is not abelian, for instance, G_1 = UT(3,p) for some odd prime number p. Take G_2 to be a nontrivial abelian 2-group. Note that G_1 is a class two Lazard Lie group and G_2 is a class one Lazard Lie group. The external direct product G_1 \times G_2 is not a Lazard Lie group.