For clarification: classical Greek has two separate (though easily confusable) verbs, viz.:
1. καθαιρῶ (from κατά + αἱρέω), an ε- contracted verb meaning to 'take down', hence 'demolish', 'put down', 'destroy', the aorist of which is usually the strong form καθεῖλον;
2. the uncontracted καθαίρω (notice the different accentuation), from, or related to, the adjective καθαρ-ός (meaning 'pure', 'clean', 'clear'), and the primary stem is the same as that of the adjective, καθαρ-; the prepositional prefix κατά has nothing to do with this. The aorist is usually ἐκάθηρα, sometimes, especially in later Greek, ἐκάθαρα.
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