Lilith came to Japan for making Tooru her duo partner. They were not on good term in the beginning but later she kissed Tooru on his cheeks and told him she is going to make him her future husband. After that their relationship is quite mutual.
It seems Sakuya is a friend who goes way back. She always teases Sakuya about her attitude towards her. Lilith is the only one who knew that Sakuya fixed her eyes at the <Absolute Duo> and also knew that Sakuya has a small world, Lilith makes a conclusion that she and him will destroy it.
Both of them are on not so good terms with each-other when it comes to Tooru. Both of them always argue when they talk about him. But there are times when they both work together for the sake of Tooru's happiness.