Can anyone help me explain this

sage: o = lattice_polytope.cross_polytope(3)
sage: F = o.face_lattice()
sage: G = F.relabel(F._elements)
sage: G[0].ambient() is o
True
sage: o = lattice_polytope.cross_polytope(3)
sage: F = o.face_lattice()
sage: G = F.relabel(F._elements)
sage: G[0].ambient() is o
False
sage: G[0].ambient() is G[1].ambient()
True

More importantly, I am interested in a work around, such that relabeling 
does not create copies.

It works once with every lattice polytope and the second time it fails. The 
only thing that helps is to quit sage and restart it.

Currently, I'm working on using CombinatorialPolyhedron to obtain the face 
lattice for lattice polytopes. It works nicely, but that the faces do not 
know the ambient polytope (exactly) anymore, which makes the doctests fail.
Somehow the current setup works, but I'm puzzled on why.

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