Ex situation

So my ex and I just recently broke up and he wants to stay friends. He cheated a couple times during the relationship but I could see by the end that he was remorseful and truly wanted to seek therapy and so I said that we could be friends. BUT it seems as though he’s been lying to me about things that don’t even need to be lied about lol. There was a girl that he was frequently messaging the last month that we were together. I feel like he emotionally cheated on me with her. He would say that they were just friends. I have no hard feelings or anger towards her, but when me and my ex talk, he constantly says that he’s not ready to be in a relationship even though I never even bring that up. He even confided in me, telling me that he was talking to “someone” consistently recently and that they basically didn’t have the same feelings.

Is it weird that I want to message the girl he was talking to to see if he asked to be in a relationship with her? I feel like I will get the truth from her and it will dictate whether he’s been lying to me. Because ultimately, I don’t want to be friends with someone who feels the need to lie to me on a regular basis. I also don’t want any arguments with him. I just want the truth so that I can give him a definitive on whether we’re friends or not.