LSJ
Ion. βασιλήϊον, τό, kingly dwelling, palace, X. Cyr. 2.4.3, etc. ; more common in pl., Hdt. 1.30, 178, Arist. Oec. 1352a11, etc.
seat of empire, capital, Plb. 3.15.3, D.S. 19.18, Str. 1.2.25.
royal treasury, Hdt. 2.149; pl., Isoc. 3.31.
tiara, diadem, LXX 2 Ki. 1.10, Roussel Cultes Égyptiens 233 (Delos, ii BC), OGI 90.45 (Rosetta), Plu. 2.358d, Porph. ap. Eus. PE 3.11, Horap. 1.15; metaph, τὸ β. τῆς εὐπρεπείας diadem of beauty, LXX Wi. 5.16. = ἅλιμος, Ps.-Dsc. 1.91; = λευκόϊον, Id. 3.123. Βασίλεια, τά, festival of Zeus Basileus, in Boeotia, IG 7.552, Sch. Pi. O. 7.153, IG 12(1).78; at Olbia, IPE 1.105.
Liddell-Scott-Jones, Greek-English Lexicon (9th ed., 1940)
TBESG
βασίλειον, -ου, τό
(< βασίλειος, which see), [in LXX for הֵיכָל ( Nam.2:6, Dan.6:18*), מַמְלָכָה ( 3Ki.3:1 14:8, 4Ki.15:19*), etc. ;]
__1. a capital city.
__2. Freq. in pl., τὰ β., a palace: Luk.7:25.
† (AS)
Translators Brief lexicon of Extended Strongs for Greek based on Abbot-Smith, A Manual Greek Lexicon of the New Testament (1922) (=AS), with corrections and adapted by Tyndale Scholars