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Injury leading to a lively second half in keen style (7)

Ross

I believe the answer is:

acutely

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'half in keen style' is the definition.
Both the definition and answer are adverbs. Maybe there's an association between them I don't understand?

'injury leading to a lively second' is the wordplay.
'injury' becomes 'cut' (cut is a kind of injury).
'leading' is a charade indicator (letters next to each other).
'lively second' becomes 'ely' (I can't justify this - if you can you should believe this answer much more).
'cut' after 'a' is 'acut'.
'acut'+'ely'='ACUTELY'

Can you help me to learn more?

(Other definitions for acutely that I've seen before include "Pointedly" , "with severity" , "In a shrewd way" , "Severely, keenly" , "Extremely" .)

I've seen this clue in the Evening Standard.
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