Our Kabbalistic sages explain that the name of each section into which the Torah is divided expresses the conceptual content of that section. The name is not simply the first words of each section, nor was it chosen arbitrarily. The name of this section is Chaiei Sara, the Life of Sarah.
However, this entire section deals with events that occurred after Sarah's death: 1) the purchase of the Cave of Machpelah to bury her, 2) the marriage of Isaac and Rebecca, which consoles her for her mother's death, 3) the marriage of Abraham and Keturah (Hagar) and the descendants of Ishmael.
If all these events occurred after Sarah's death, why is the parashah called "The Life of Sarah"? This signifies a profound connection between the events recounted in this parashah and Sarah's actual life.
The true expression of life is when it is eternal. This is possible only when one is connected to God, the source of life and eternity.
Life does not end with death, but rather continues through the person's descendants and the life that unfolds in the world. From this perspective, events following death specifically testify to the nature of life, because through them one can assess the essence of a person's existence.
This is the meaning of the words of our sages, "Jacob, our patriarch, did not die; if his seed is alive, he is alive." It means that when his seed continues on its path, we can be certain that "he is alive.".
In this way, we will understand one of the events recounted in this section: we see Rivka, who continues the path of Sara, to the point of generating the same miracles that occurred in Sara's life, as the Midrash tells us, that in Sara's tent there was a candle lit from one Friday to the next, that there was a special blessing over the bread dough and that there was a cloud always protecting the tent.
To understand the other two (the purchase of the cave and the descendants of Abraham) we must first understand the difference between Abraham and Sarah.
Abraham was not only the father of Isaac, but also of Ishmael. From Abraham descended not only the Jewish people, but also other nations. In contrast, Sarah was the mother of Isaac only.
The same difference is expressed in Abraham and Sarah's service to God. Abraham spread the knowledge of God to all people, even the Arabs of that time who knelt at his feet. In the words of Kabbalah, Abraham projected divine energy to all of humanity. Sarah, on the other hand, from the birth of Isaac, only desired to bestow divine energy upon Isaac, to the point of expelling Ishmael and Hagar from Abraham's house.
This concept of the transmission of divine energy to everyone (Avraham) or only to Itzjak (Sara) is the essence of the explanation of the other two events of the parashah.
The significance of the Cave of Machpelah lay in the fact that Adam and Eve, the parents of all humanity, were buried there. This apparently relates to all of humanity and not just the Jewish people. However, Abraham purchased the cave to bury Sarah, and indeed, the patriarchs and matriarchs of the Jewish people are buried there, indicating that the cave belongs exclusively to the Jewish people.
Later, the Torah tells us about Abraham's descendants and his marriage to Keturah. In this account, the Torah tells us that Abraham gave everything he owned to Isaac, while to his other sons he gave only gifts. Even when the Torah refers to Ishmael, he is described as "the son of Hagar, the Egyptian maidservant of Sarah," meaning that there was no comparison between Isaac, the son of Sarah, and Ishmael, the son of Sarah's maidservant.
This leads us to understand that the events that occurred after Sara's death were the culmination of her life's work. Sara's path in life, prioritizing Itzjak above the rest of the world, was the right one, and that is why this section is called Jaiei Sara, Sara's Life, because all the events described indicate how Sara's path continued even after her death.
Extracted from Likutei Sijot, vol 15, p. 145