John Chrysostom

"And forasmuch as not without the taking of an oath." Thou seest that the matter of the oath becomes necessary for him here. Accordingly for this reason he previously treated much hereon, how that God swore; and swore for the sake of our fuller assurance.

Homily on Hebrews 13 ↗

Theophylact of Ohrid

Here is another distinction of the new Priest from the ancient ones and of one covenant from the other. For the priesthood of Christ was promised not simply, but with an oath, so that the word of God would be fully believed, just as above, for greater persuasiveness, he said that God swore to Abraham (Heb. 6:13).

Commentary on Hebrews ↗

Oecumenius

"because the matter was not done without an oath." And so much, he says, the hope has been greatly improved, insofar as Christ has become a priest without an oath. For the oath indicates the unchangeable nature of his being a high priest forever.

The Pseudo-Oecumenian Catena on Hebrews ↗

Thomas Aquinas

364. – Having proved by the authority of the Psalmist that the priesthood of Christ is preferred to the Levitical and does away with it, the Apostle now proves the same thing by two other authorities: first, from the fact that he says, 'The Lord has sworn'; secondly, that he says, 'You are a priest' (v. 23).

Commentary on Hebrews ↗