That thou shouldest enter - לעבר leaber, to pass through, that is, between the separated parts of the covenant sacrifice. See Genesis 15:18 (note).
And into his oath - Thus we find that in a covenant were these seven particulars:
1. The parties about to contract were considered as being hitherto separated.
2. They now agree to enter into a state of close and permanent amity.
3. They meet together in a solemn manner for this purpose.
4. A sacrifice is offered to God on the occasion, for the whole is a religious act.
5. The victim is separated exactly into two equal parts, the separation being in the direction of the spine; and those parts are laid opposite to each other, sufficient room being allowed for the contracting parties to pass between them.
6. The contracting parties meet in the victim, and the conditions of the covenant by which they are to be mutually bound are recited.
7. An oath is taken by these parties that they shall punctually and faithfully perform their respective conditions, and thus the covenant is made and ratified. See Jeremiah 34:18, Jeremiah 34:19, and the notes on Genesis 6:18 (note); Genesis 15:18 (note); Exodus 29:45 (note); Leviticus 26 (note).

That thou shouldest (g) enter into covenant with the LORD thy God, and into his oath, which the LORD thy God maketh with thee this day:
(g) Alluding to them, that when they made a sure covenant, divided a beast in two, and past between the parts divided, (Genesis 15:10).

That thou shouldest enter into covenant with the Lord thy God,.... That is, they were all to appear and stand in this order before the Lord, that they might solemnly avouch him to be their God, and hear him declaring them to be his people, and the many promises and prophecies of good things he should deliver to them, as well as threatenings of wrath and vengeance in case of disobedience to him: or "that thou shouldest pass" (e): which some think is an allusion to the manner of making covenants, by slaying a creature, and cutting it in pieces, and passing between them, as in Jeremiah 34:18; so Jarchi and Aben Ezra:
and into his oath; annexed to his covenant and promise, to show the immutability and certain fulfilment of it on his part; and may signify not only the oath he swore that they should be his people, but the oath he gave them, and they took, that he should be their God:
which the Lord thy God maketh with thee this day; which refers both to the covenant and the oath, or the covenant confirmed by an oath, even the covenant now made in the plains of Moab, distinct from that at Horeb or Sinai.
(e) "ut transeas", V. L. Tigurine version, Munster, Vatablus, Pagniuns, Cocceius; "ad transeundum", Montanus.

"That thou shouldest enter into the covenant of the Lord thy God, and the engagement on oath, which the Lord thy God concludeth with thee to-day." עבר with בּ, as in Job 33:28, "to enter into," expresses entire entrance, which goes completely through the territory entered, and is more emphatic than בברית בּוא (2-Chronicles 15:12). "Into the oath:" the covenant confirmed with an oath, covenants being always accompanied with oaths (vid., Genesis 26:28).

Into covenant and into his oath - Into covenant, confirmed by a solemn oath.