We still aren't quite to a point where even allianced airlines have IT systems that talk to each other reliably. I'm just back from a trip where the LH/EW part of my reservation had my first/last names in the reversed fields, according to the helpful agent at the LH Senator Lounge in DUS. Empirically, there was clearly something wrong with that record because I had hiccoughs that required minor manual intervention at every step of the trip, despite the fact that I've flown similar itineraries without issue that were also booked on UA's web site.

So, book LH on UA with some caution. Not everything will work as smoothly as if you book on LH directly. You may, for example, lose the ability to select your seating, or do on-line check-in. Or maybe not. The uncertainty is evidence that robust coordination is a moving target for these two IT teams.