Besides the cakes, he shall offer for his offering leavened bread with the sacrifice of thanksgiving of his peace offerings.

Wesley's Notes

7:13 Leavened bread - Because this was a sacrifice of another kind than those in which leaven was forbidden, this being a sacrifice of thanksgiving for God's blessings, among which leavened bread was one. Leaven indeed was universally forbidden, Lev 2:11. But that prohibition concerned only things offered and burnt upon the altar, which this bread was not.

Scofield Reference Notes

[2] {leaven]

The use of leaven here is significant. Peace with God is something which the believer shares with God. Christ is our peace-offering Eph 2:13. Any thanksgiving for peace must, first of all, present Him. In verse 12 we have this, in type, and Song leaven is excluded. In verse 13 it is the offerer who gives thanks for his participation in the peace, and Song leaven fitly signifies, that though having peace with God through the work of another, there is still evil in him. This is illustrated in Amos 4:5 where the evil in Israel is before God.

Matthew Henry's Concise Commentary

7:11-27 As to the peace-offerings, in the expression of their sense of mercy, God left them more at liberty, than in the expression of their sense of sin; that their sacrifices, being free-will offerings, might be the more acceptable, while, by obliging them to bring the sacrifices of atonement, God shows the necessity of the great Propitiation. The main reason why blood was forbidden of old, was because the Lord had appointed blood for an atonement. This use, being figurative, had its end in Christ, who by his death and blood-shedding caused the sacrifices to cease. Therefore this law is not now in force on believers.