I was looking at example sentences for 「気づく」 to determine which particle it
uses and I found that it takes 「に」. For example: 「彼はそれ **に** 間もなく気づくだろう」.
However, I also found these sentences: 「私は彼が有名人 **だと** 気づく」 and 「自分が一人
**であるのに** 気づく」.

Firstly, I'm struggling to analyse 「一人であるのに気づく」; 一人 (noun) + であるの
(nominaliser) + に気づく is this correct? I thought that you only used である with こと
(e.g. 先生であること), but you use な with の (e.g. 先生なの).

Secondly, I have no clue what 「だと」 means in the other example. Again, it looks
like a nominalised sentence to me, but I'm not sure how 「だと」 is used. Even if
it was nominalised, why is と used and not に?
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question
grammar particles nominalization
What is the difference between 「だと」 and 「であること・なの」
There are a few (probably more than I'm thinking of) cases where に and と can
be used interchangeably, such as に比べて/と比べて, and I think that's the case with
気づく also.

私は彼が有名人だと気づく

There's an implicit nominalization here: 彼が有名人だ (the fact that he is famous)
と気づく (I realize). It's similar (probably related) to when you use と as the
quoting particle without using quotation marks.

As for 一人であること, I don't think the rule you learned about when the の
nominalizer is required is correct. 一人なのに気づく sounds like you're saying, "I
realize it **despite the fact** that I am alone."

**Edit:** Just thought of a more obvious analogy. I trust that nothing seems
odd about the だと combination in this sentence:

有名人だと思う。
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answer
Like "to realize" can take either an object or a clause and you can say
"realize it" and "realize that something is something" respectively, 気づく can
be combined with either an indirect object or a quotation clause and can be
said as "(indirect object) に気づく" and "(clause)と 気づく".

In order to nominalize the clause, you can use abstract noun こと or pronoun の
with the verb in the clause in the attributive form, that of the copula is
（で）ある or な. Once you nominalize the clause, you can regard it as an indirect
object for the structure of …に気づく.

In summary, you can change …だと気づく into {…である / …な} + {こと / の} に 気づく.
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answer