Thematic Significance of the Death of Cordelia in King Lear

I have to write an essay about how the death of Cordelia highlights the meaning of the play. I am trying to look for different arguments, and I hope to expand on 4 points, with two quotes each.

First Point: idea of an unjust world (the fact that Cordelia was the only one that truly loved King Lear)
Second Point: the death acts as King Lear's personal sacrifice (but I am not sure how to explain this one)

After those two points, I am to sure what else I can use. I have been thinking about the theme of betrayal (Lear's betrayal towards the kingdom, Goneril and Regan), the motif of the chain of being (Lear is degraded to madness, chaos), self recognition, and wheel of fortune. Can someone please help me expand on these points?

it is too short

Perhaps, you could insert characters background to make you essay longer.