Albert Barnes Commentary Leviticus 10:6-7

"And Moses said unto Aaron, and unto Eleazar and unto Ithamar, his sons, Let not the hair of your heads go loose, neither rend your clothes; that ye die not, and that he be not wroth with all the congregation: but let your brethren, the whole house of Israel, bewail the burning which Jehovah hath kindled. And ye shall not go out from the door of the tent of meeting, lest ye die; for the anointing oil of Jehovah is upon you. And they did according to the word of Moses." — Leviticus 10:6-7 (ASV)

Aaron and his two surviving sons are forbidden to show the accustomed signs of mourning, or to leave the court of the tabernacle to attend the funeral, because, due to their office, they were especially responsible as consecrated priests for outwardly maintaining the honor of Yahweh. They were to bear visible testimony to the righteousness of the punishment of Nadab and Abihu. The people, on the other hand, not formally standing so near to Yahweh, were permitted to “bewail” as an acknowledgment that the nation shared in the sin of its priests (Compare to 1 Corinthians 12:26).

In Leviticus 10:6, the command Uncover not your heads—or, “set free... let go loose”—explains a custom. It was customary to let the hair grow long and fall loosely over the head and face (Leviticus 13:45; 2 Samuel 15:30; 2 Samuel 19:4); the substance of the command, therefore, was that they should not let their hair go disheveled. Ripping clothes in front to lay the breast open was one of the most common manifestations of grief (Genesis 44:13; 2 Samuel 1:11; Job 1:20; Joel 2:13, etc.). The garments, as well as the persons of the priests, were consecrated; this appears to be the reason for the prohibition of these ordinary signs of mourning .

The phrase Lest you die, also from Leviticus 10:6, is further elucidated by a comparison (see the note on Exodus 28:35).

Regarding Leviticus 10:7, the phrase The anointing oil ... is upon you (referencing instructions in Leviticus 8:12; Leviticus 8:30) signifies a profound truth. The holy oil, as the symbol of the Holy Spirit—the Spirit of life and immortality and joy—was the sign of the priests being brought near to Yahweh. It was therefore, by its meaning, connected both with the general law that forbade the high priest ever to put on signs of mourning on account of death (Leviticus 21:10–12) and with the special reason for the prohibition on this occasion.