Nomenclature for stereoisomer example possible wrong? I understand the principal of the Cis-Trans naming system (at least I think so), until I saw this example in my learning material. My teacher doesn't know either so I thought I would post it here.

This is the molecule:


There are two ethyl groups both on the bottom side of the double bond, so it could be cis right?


Apparently not, the book says this is the answer:


Could someone please explain the reasoning behind this? This particular example is unusual because there are three ethyl groups attached to the double bond so I cannot determine which one should be the parent chain. The book doesn't explain.
Unless I'm very much mistaken, this molecule should not be labelled with any stereochemical descriptors, whether they be relative (cis/trans) or absolute (E/Z). The double bond has three identical ethyl substituents, therefore there is no stereoisomerism possible. By way of explanation, imagine if the ethyl group and the hydrogen on carbon #4 were interchanged: the resulting molecule would be completely identical to the one pictured. This is always the case when an alkene bears two identical substituents on the same carbon atom.

To correctly use cis/trans or E/Z nomenclature, it is necessary that there not be two identical groups on the same carbon of the alkene; if there are, the labels are meaningless, since interchanging the two groups results in an identical molecule.