This is a followup to my post on meta.SE.
The filler text:
Apparently, this user prefers to keep an air of mystery about them.
sounds odd to my ear. I understand that "them" is being used as a singular, third-person gender neutral pronoun, but I don't think that's the issue. The analogous gender-specific constructions:
Apparently, this user prefers to keep an air of mystery about him.
Apparently, this user prefers to keep an air of mystery about her.
seem to be archaic and awkward, more naturally replaced by:
Apparently, this user prefers to keep an air of mystery about himself.
Apparently, this user prefers to keep an air of mystery about herself.
While it's true that "an air of X about them" is at least a moderately-common construction in English, the construction "keep an air of X" is less common, so I don't think that the example of "There was an air of secrecy about them" is informative in this case.
I think the construction here is "to keep an X about Y" is the one that is important, and in the example of:
They keep their friends close to themselves and their enemies closer.
They keep their friends close to them and their enemies closer.
I'd say that the former is more correct than the latter.
I have no references for this, however, and I'm not sure where I would look. Is this a question of style, or grammar? Is there any authoritative reference for this? Is there an informative Google ngram search that would lean one way or the other?