In short, we know of the death of Martin Septim, as we witnessed it first hand. We know both sons of Uriel Vll were assassinated, and that Uriel himself was killed. Furthermore, we all know that the Septim bloodline truly ended with the death of Hjolti Half-Beard of Aclaire.(Figuratively)

Now, when, in Septim history, could there have been any remaining bloodlines that were, I don't know, disinherited?

Empress Katariah had a son named Cassynder. He was half-Breton half-Dunmer. At the time of Katariah's rule, many would say members of the council made things slightly unpleasant for the Dark Elf empress. Which, led to her constant travelling through Tamriel. In fact, she travelled Tamriel more than any other ruler, aside from Tiber. This lead to her death whilst in a small skirmish in Black Marsh. Anyway, after his mother's death, Cassynder was pulled from Wayrest, it's former king, and crowned Cassynder Septim. During his rule in Wayrest, Cassynder adopted a brother into the Septim family by the name of Uriel Lariat. Having already been ill from old age, Uriel succeeded his step-brother as Uriel Septim the lV.

Once Cassynder had left Wayrest, Uriel took his place, but now, having been summoned to take the throne as Emperor, he too had to abandon Wayrest to a distant and irrelevant other. Now, Uriel Lariat, now Uriel Septim lV, ruled for 43 years in Cyrodiil. Of course, the Council had this tender and practically powerless king under their thumb. So, Uriel's influence was limited. Uriel had a son named Andorak, who the Council, in their drunken power, disinherited to crown Cephorus ll. It was said that those loyal to Uriel and Andorak went into upheaval until the Council gave Andorak Shornhelm, crowning him the king of the town/city. It was also written that Andorak's offspring still rule there.

So, in conclusion, I believe that the children of his children are suitable recipients of the crown and throne as Emporer or Empress of Tamriel.

(If my information should prove to be inaccurate or incorrect, sorry for your wasted time.)

The throne belongs to whoever has the power to hold it. Just like old Tiber once took it.
Blood and heritage only matter because they might provide you with followers amongst the nobility.

And it also required the Emperor to be recognized by the Amulet of Kings. But there's no more amulet, and no more dragonfires. It'll be interesting to see where it goes.

Lord Dren: I hadn't realized that. Nice find.

Though I imagine there will be others too, the Septim family has a rather broad tree and the crown has jumped from one branch to another before.