Who do we keep an air of mystery about?

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?

First off, The usage of about is different here. [Oxford]

You think it's used in sense 1

On the subject of; concerning:

When in fact it's used in sense 3

(chiefly British) Used to express location in a particular place:

rugs were strewn about the hall

They're trying to say users keep an air of mystery around them

It's not used in this sense in AmE, maybe that caused the confusion.

See the 'Which Word?' segment in this Oxford entry

About the reflexive pronoun:

You are quite right in stating that even in this sense, subject and object are the same and so the reflexive pronoun is applicable. It isn't used for two reasons:

• "Air of mystery about him" is an old idiomatic way of putting it

• Using the reflexive emphasises the object, and so it is sometimes not used when this emphasis isn't desired, especially if the subject and object are unambiguously the same.

Compare:

• "He always keeps his diary with himself" (= he doesn't give it to anyone else.)
• "He always keeps his diary with him" (= he carries it around all day)

In "Apparently, this user prefers to keep an air of mystery about them.", the emphasis is desired on mystery, and it is highly unlikely anyone would interpret it to mean the user keeps the air about someone else.