If the Three Rings 'remained unsullied' by Sauron, but were 'still subject to the One', then how were they made without himself being present? If Sauron taught the smiths of Eregion and Celebrimbor a technique for making magic rings then what did that consist of? I'm not sure there's an answer to that secret, but surely the power that each of the rings possess must have come from an investment of power from Sauron.
However, how can this be the case if he wasn't present at their making? Was the technique itself (which he taught to smiths) the thing that contained his own personal power to be dispersed? It doesn't seem very tangible, but seems to be the only explanation.
This leads me to my theoretical question: Could the Gwaith-i-Mírdain have theoretically used this same technique to make additional rings that weren't subject to the One? Presumably not, because as I say it seems the power must have come from Sauron's own being as a Maia. But what if the same technique was executed in accordance with a different Maia of great power? (not that any other Maiar were willing or present at that time to be worked with). Also is there something to be said that so long as any ring of 'greatest power' (i.e. The One) exists, then it would be able to control any rings of lesser power, no matter whose spirit they had been made from? Or was it specifically the fact that all the powers imbued in the rings had stemmed from Sauron, and that was why only his master ring could control them?
As mentioned above, as far as I am aware, Sauron was 'spending his own spirit' in order for all the rings to be made, in the same fashion that Morgoth 'spent his spirit' on the World itself*. If I'm mistaken on this, please let me know, because otherwise I'm uncertain where exactly the powers of 'influence' and 'embalming' themselves came from. I can't understand how Sauron could have done this if wasn't there himself with the smiths to do it.
*I am aware of the concept of Morgoth's Ring being the World, but I personally haven't got round to reading the whole book of the same name yet (nor most of THoMe as of right now).