Adam Clarke on Hebrews 13:12

That he might sanctify the people - That he might consecrate them to God, and make an atonement for their sins, he suffered without the gate at Jerusalem, as the sin-offering was consumed without the camp when the tabernacle abode in the wilderness. Perhaps all this was typical of the abolition of the Jewish sacrifices, and the termination of the whole Levitical system of worship. He left the city, denounced its final destruction, and abandoned it to its fate; and suffered without the gate to bring the Gentiles to God.