LSJ
and indeed, and further, freq. in Hom. with one or more words between, Il. 1.426, al. ; καὶ σύ τοι E. Med. 344; καὶ τἆλλά τοι X. Cyr. 7.3.10; once in Hom. as one word, Il. 13.267. after Hom. usu., and yet, to mark an objection introduced by the speaker himself, freq. in Rhetorical questions, καίτοι τί φημι ; A. Pr. 101; κ. τί φωνῶ ; S. OC 1132, cf. Isoc. 4.99, etc. ; without a question, κ. φύγοιμ’ ἄν E. Cyc. 480; κ. καὶ τοῦτο… D. 4.12, 18.122; strengthd., καίτοι γ’ Ar. Ach. 611, E. Fr. 953.10, X. Mem. 1.2.3, Ph. 1.274, etc. ; mostly separated, καίτοι… γε E. Or. 77, Ar. Ra. 43, X. Mem. 3.12.7, etc. (καίτοι is f.l. in A. Eu. 849); so καίτοι περ v.l. in Hdt. 8.53. with a participle, much like καίπερ, Simon. 5.9, Ar. Ec. 159, Pl. R. 511d, Plb. 22.8.13, Phld. Ir. p. 22 W., Luc. Alex. 3; once in the Att. Oratt., Lys. 31.34; also καίτοι γε διαχλευάζων Pl. Ax. 364b.
Liddell-Scott-Jones, Greek-English Lexicon (9th ed., 1940)