I don't think it's actually said anywhere.

Fake info!?

Only thing close enough is his Mugshot which was taken December 7th, 1874

It's only ever stated that Lyle died in-front of Arthur. It's never that he was hanged.

I made a discussion about this. I made the theory that Arthur’s father was hanged. But what I said was a theory, nothing else. And no, I don’t believe it was ever stated anywhere in the game (unless I missed something), so it shouldn’t be on either of their articles