Lazard Lie property is not finite direct product-closed

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.