Right radical 几 - the indexing component. Pictures a low stool or table used as a graphic anchor. In modern usage 凯 carries the triumphal-music sense (凯旋, 凯歌) and serves as a popular character in transliterated names: 凯文 Kevin, 凯特 Kate, 凯瑟琳 Catherine.
Left 岂 supplies the sound - qǐ drifted to kǎi via initial fronting. Traditional 凱 used the fuller 豈, originally a pictograph of a war drum mounted on a stand and struck during triumphal processions. Same phonetic in 楷 (model script) and 揩 (wipe).
