Brothers: one did for the other an Alice band, bizarrely (4,3,4)

I believe the answer is:

cain and abel

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'brothers' is the definition.
(I've seen this before)

'an alice band bizarrely' is the wordplay.
'bizarrely' is an anagram indicator.
'an'+'alice'+'band'='analiceband'
'analiceband' is an anagram of 'CAIN AND ABEL'.

'one did for the other' is the link.
This may not be right. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for cain and abel that I've seen before include "Biblical brothers" , "Adam and Eve's sons" .)

I've seen this clue in The Times.

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