Summary: They play chess.
It was just chess.
The first time they played, John's team were late. Elizabeth was pacing her office when Steven saw the board and challenged her to a game. It took her mind off the wait and the good news came just after he checkmated her. Smiling, she asked for a rematch.
It became a ritual. Whenever he was in Atlantis she challenged him for a game. Sometimes she won; often he did. The games helped him forget his bitterness at not being able to stay in Atlantis; it helped her forget the Earth she still missed.
Despite the time they spent together, they only ever spoke of work, but that wasn't always a safe topic as they vied for power. They couldn't even agree on how to handle the Wraith. Most games were played in silence and chess became their conversation.
He gambled with pawns, she tempted him with her queen. She backed him into a corner, he found a way out. They knew each other's moves; they learned new ones to surprise each other. When she kissed him neither of them said anything. When he was next on Atlantis, he kissed her.
It wasn't just chess any more.