What case of 'I' should be used in a Participle Clause? [duplicate]

Which of the following sentences is grammatically correct?

1. I being not at home, my friend left a message.

2. Me being not at home, my friend left a message.

3. My being not at home, my friend left a message.

I think that the third sentence can be easily ruled out since My being not at home would be a gerund, and we need a participle here.

I have seen sentences like Him being dead, there was no one remaining to challenge her. and I think they are grammatically correct. Following this trend of using the objective case, I believe that the second sentence would be the correct one, but my teacher says that it should be the first one (probably because the second one does have an informal feel to it).

So, which one of them is the correct one, and why?

[1] [I being not at home], my friend left a message.

[2] [Me being not at home], my friend left a message.

[3] * [My being not at home], my friend left a message.

The bracketed expressions are gerund-participial clauses in adjunct function.

In such constructions, genitive subjects are not permitted at all: the choice is between nominative [2] and accusative [1], with [3] being ungrammatical.

Incidentally, it would be more natural to put the negative marker "not" before the verb rather than after it: "I/me not being at home".

Grammatically both 1 and 2 are correct. The difference is not a matter of grammar but of style. It's just that 1 is more formal while 2 is less formal.