The answer by Imprfectluck at this question made me wonder what would happen to people whose name was written down in the details of another persons death. For example: John Doe dies after his date with Jane Doe. The foremost basic rule of the Death Note states that
The human whose name is written in this note shall die. So will Jane Doe die?
We additionally have rule LIV that states that if you write the cause and details of death on a different page, you need to be thinking of the original victim and would not be able to think both of John and Jane at the same time. This would indicate that you have to choose what will happen. Either you think of John and the details happen, either you think of Jane and she dies. Does a similar rule exist when writing on the same page? Or is writing someone's name in the details of another's death just an exception to that first rule?
In order to make the Death Note take effect, the victim's name must be written on the same page, but the cause of death and situation around the death can be described in other pages of Death Note. This will work as long as the person that writes in the Death Note keeps the specific victims name in mind when writing the cause and situation of death.