Yep, that's why she's so obsessed with being loved.
That sister is the daughter of her father's concubine. Li's mother was so jealous of that concubine because it was clear that Li's dad loved her more. Plus, when she was young she could see how much of a happy family they were. She couldn't have that and instead poured all her energy into making the Emperor hers.
But alas, it was all for nothing.
Not just that, FL's mother had a weak body to begin with but had to give the father a child. Having FL worsent her health and so the father brought in his "beloved" comcubine to try for a heir and locked the mother away in a seperate building.. all under the guise that "it was better for her health". But the father loved the comcubine so the maids desided to abandone FL and her mother who was sick so FL grew up looking at her well feed well dressed loved father, comcubine and sister. FL had to fight the maids to get just enought food to not starve and then her mother died.
The comcubine became the second wife (married less than a month after FL's mother was discoverd dead). Comcubine forced the father into being a father for FL by saying "you owe her for the years of neglect".
As FL grew older the father chose to abandone the sister as a way of "making it up to Fl for the neglect" aka FL was the daughter raised with love (Outside the family ppl only know the father to have one child..)
SHE HAS A SISTER??