In an elect or chosen manner; as one who is selected or chosen.
From 'elect' (as an adjective meaning 'chosen' or 'select') plus the adverb suffix '-ly'. In religious contexts, 'elect' refers to those chosen by God, making 'electly' a theology-tinged adverb.
This is an ultra-rare word that would have appeared in religious texts—'the saved shall live electly in grace.' It's vanished because 'elect' stopped being used as an adjective in modern English. Most people wouldn't recognize it if they saw it!
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