Alexis might confront Beckett directly. She could say something like 'I think you're cheating on me. What's going on?' This is a very straightforward approach.
I don't know who Alexis is in this story without reading it. But it's likely that Alexis is either the main character or an important figure in the events that unfold in the 'creepy subway' setting.
Abigail and John Adams had a deep and abiding love. They were pen pals when John was away on political business. Their letters were filled with affection, thoughts, and mutual respect. Abigail was an intelligent and strong - willed woman, and John admired her greatly. They supported each other through John's political career, with Abigail often giving him sage advice. Their relationship was a partnership in every sense, and they remained devoted to each other throughout their lives.
A lawyer could be an important character. This lawyer takes on Alexis's case, believing in their innocence. The lawyer has to battle a corrupt legal system or powerful opponents who want to see Alexis locked up for good. The fanfic could show the lawyer's strategies and how they build a case in Alexis's favor.
I'm not sure specifically as I don't have prior knowledge of Gerard Adams' story. It could be about his personal achievements, his life journey, or his experiences in a particular field.
Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
One key element was their intellectual connection. They could exchange ideas freely. Another was trust. John trusted Abigail to handle things at home while he was away. Also, mutual admiration played a big role. Abigail admired John's political pursuits, and John admired Abigail's intelligence.
I'm not sure who Alexis Rhodes specifically is in this context as the name is rather common. There could be many fictional or real people named Alexis Rhodes. Maybe it's a character from a particular book or a personal account that I'm not familiar with.