As much as Lionel wanted to find out about Desire's origins, he wasn't able to do it.
After all, not even Desire knows much about himself. There's no way Lionel would be able to find things out just because he knew more things.
It might be possible if Lionel would use everything he has to find out Desire's origins.
However, not even that could guarantee him that he would know about Desire more.
That was why Lionel stopped thinking about it and just observed Desire.
After all, although Desire has indeed gone through the first stage of Refreshing, Death, it is not complete.
The body must be more stimulated than it is now, and Desire should have the ability to 'wake up' whenever he wanted.
After all, this state of his could be called the borderline of death and life.
However, Lionel was sure that Desire didn't know how to wake up from this.
That was why it was dangerous to let things go on like this.