[JJM Ballantyne]

The simplest way to approach this is to repeat the following several times:

"there is no such thing as a split infinitive"

To illustrate, "to argue" is not an infinitive, "argue" is; "to argue" is the construction "prepositional marker 'to' + 'infinitive' (base verb form)." We know this because the prepositional marker "to" is not always required with the infinitive:

Jim does argue a lot

I will argue your case later

So, to truly split (ha!) the infinitive, we'd have to come up with something like:

I'm too tired to arreallygue with you

English does not allow us to do this sort of infixing.

So where does that leave us?

It leaves us without the daffy and spurious concept of the "split infinitive." It means that the principal guide in choosing to place another word between the marker "to" and the "infinitive" is whether it "sounds right" stylistically and the alliteration is effective.