Since the beginning, Ishmael and Isaac seem to be connected, even in the way their birth is described:

So Hagar bore Abram a son; and AbramFor Sarah conceived and bore Abraham a
named  his  son,  whom  Hagar  bore,son in his old age, at the set time of which
Ishmael. Abram was eighty-six years oldGod had spoken to him. And Abraham
when Hagar bore Ishmael to Abram. (Gencalled the name of his son who was born
16:15, 16)to him—whom Sarah bore to him—Isaac.
 (Gen 21:2, 3)

Except for the somewhat ambiguous text of Genesis 21:9, Ishmael and Isaac do not seem to have personal difficulties with each other. Their mothers are the ones who do not get along. In verse 9 there is a word play for the word “laughter” to describe Ishmael “laughing” (ESV), “playing” (NRSV) with Isaac or “mocking”(ASV), or “scoffing” (NKJV) him, depending on the translation. In the original language the expression sounds like Ishmael was “Isaacing” with Isaac. The original word for Ishmael’s “laughing” (צחק) actually points to the meaning of Isaac’s name. The same word is used to describe the laughing of Abraham (Gen 17:17), Sarah (Gen 18:12, 13, 15), and Isaac himself (Gen 26:8).

In Genesis 21 Isaac and Ishmael are separated because of family conflict. However, their connection seems to continue on through the parallelism found in Scripture.

At a certain moment in his adult life, Isaac moves into the proximity of Beer-Laai-Roi (Gen 24:62), a well named by Hagar as gratitude for the vision of God and the promises given to her son Ishmael who is also Isaac’s brother. The story says that Isaac had come from this well and was now meditating or praying. The well Beer-Laai-Roi is now connected with God’s revelation to Hagar, the promises of God to his brother Ishmael, and above all points to the fact that God is one who hears, sees, and knows. The narrative seems to connect Isaac to this reality. As God heard Hagar by the well, so Isaac desires that God would hear and guide him in his future marriage. He, too, searches for the God who hears. He, too, will be comforted. Here there is a connection between Isaac and Ishmael’s mother, Hagar.

Another event seems to cast light on Isaac’s relationship with Ishmael. The blessing Esau had lost to Jacob, seemed to be connected with Isaac’s command “not [to] take a wife from the daughters of Canaan” and Jacob’s obedience to the command.

Esau saw that Isaac had blessed Jacob and sent him away to Padan Aram to take himself a wife from there, and that as he blessed him he gave him a charge, saying, “You shall not take a wife from the daughters of Canaan,” and that Jacob had obeyed his father and his mother and had gone to Padan Aram. Also Esau saw that the daughters of Canaan did not please his father Isaac. So Esau went to Ishmael and took Mahalath the daughter of Ishmael, Abraham’s son, the sister of Nebajoth, to be his wife in addition to the wives he had. (Gen 28:6-9, NKJV, emphasis mine)

Isaac and Ishmael meet again when their father died. (In a similar way, Esau and Jacob meet for their father’s funeral, which seems to confirm their reconciliation.) Their meeting is noteworthy. Only Ishmael and Isaac meet here. The other sons of Abraham are not mentioned. Only Isaac and Ishmael are called “sons” of Abraham in Genesis. The other sons are not called Abraham’s but Keturah’s sons (Gen 25:4; 28:9; 1 Chr 1:32).