C. F. Keil and Franz Delitzsch on 1 Kings 2:34

1Ki 2:34

Benaiah went up (ויּעל), inasmuch as the altar by the ark of the covenant stood higher up Mount Zion than Solomon’s house. Joab was buried “in his house” (i.e., in the tomb prepared in his house, either in the court or in the garden: cf. 1Sa 25:1), “in the desert,” probably the wilderness of Judah, as Joab’s mother was a step-sister of David, and therefore probably dwelt in the neighbourhood of Bethlehem.