https://en.wikipedia.org/wiki/Antonia_Fortress

Holy Vestments lists priestly garb, Armor of God lists soldier's uniform

Antonia was basically the sacristy, making it an integral part of the temple and modern churches... see Priest-Soldier parallel

priest/soldiers 'hold fast/stand firm'

• Ephesians 6:13  (histemi, G2476) and Hebrews 4:14 (krateo, G2902). Hebrews relies on order of Mechelzidek, not Levite/Aaron. Also, if no sacrifices are needed, then neither are holy vestments...

Acts 21:31 claims one cohort, but can Luke be trusted to know military units? also, a tribune would have nominally had two cohorts, not just one.

πτερύγιον

Josephus calls Antonia an akropolis (citadel) for the arkhiereus (high priest) where the stole (vestments) for the high priest were kept. Antiquities of the Jews  and Book XV, chapter 11.4 (see below) and again Book XVIII,  Chapter 4.3

Κατὰ δὲ τὴν βόρειον πλευρὰν ἀκρόπολις ἐγγώνιος εὐερκὴς ἐτετείχιστο διάφορος ἐχυρότητι. ταύτην οἱ πρὸ Ἡρώδου τοῦ Ἀσαμωναίων γένους βασιλεῖς καὶ ἀρχιερεῖς ᾠκοδόμησαν καὶ βᾶριν ἐκάλεσαν, ὡς ἐκεῖ τὴν ἱερατικὴν αὐτοῖς ἀποκεῖσθαι στολήν, ἣν ὅταν δέῃ θύειν τότε μόνον ὁ ἀρχιερεὺς ἀμφιέννυται.

Josephus Wars,

Πυργοειδὴς δὲ οὖσα τὸ πᾶν σχῆμα κατὰ γωνίαν τέσσαρσιν ἑτέροις διείληπτο πύργοις, ὧν οἱ μὲν ἄλλοι πεντήκοντα τὸ ὕψος, ὁ δ' ἐπὶ τῇ μεσημβρινῇ καὶ κατὰ ἀνατολὴν γωνίᾳ κείμενος ἑβδομήκοντα πηχῶν ἦν, ὡς καθορᾶν ὅλον ἀπ' αὐτοῦ τὸ ἱερόν.

• "Since the entire structure was that of a tower, it contained four other distinct towers at its four corners, three of which the others were fifty feet high but the one on the southeast corner was seventy feet high, commanding a view of the whole temple."

Matthew 4

Luke 4

πτερύγιον (pterygion, G4419) pinnacle

πύργος (pyrgos, G4444) tower