וַיָּ֤שֶׂם הָעֶ֙בֶד֙ אֶת־יָד֔וֹ תַּ֛חַת יֶ֥רֶךְ אַבְרָהָ֖ם אֲדֹנָ֑יו וַיִּשָּׁ֣בַֽע ל֔וֹ עַל־הַדָּבָ֖ר הַזֶּֽה׃
And the servant put his hand under the thigh of Abraham his master, and swore to him concerning this matter.
Rashbam on Genesis
על הדבר הזה, concerning the condition stipulated by Avraham.
Sforno on Genesis
וישבע לו על הדבר הזה, he accepted all of Avraham’s conditions as part of his oath.
Or HaChaim on Genesis
וישבע לו על הדבר. He swore to him concerning the matter. An absolute oath concerning a Canaanite girl, and a conditional oath concerning the need to find the girl in Abraham's birthplace should the girl refuse to move to the land of Canaan.