Abelian Group - Commutative Group, which means that it can be swapped without changing meaning or value, which can only happen if it is not having any affinity or attachment.
Sanskrit 'bil' means 'to cleave' and 'to split' - one can not 'attach' unless one cleaves something else or splits the object of attachment.
In Sanskrit and Hindi, "Bel" is the name for the Creeper Plant, which needs 'cling' to the host, usually wall or a pole, to climb. In Sanskrit, vowels get graded to different levels, so "i" ie connected to "ii", "e", and "ae" - as well as 'y', 'ay', 'aya' and 'ya', which is opposite of vowel gradation, and is called retro-gradation, and in Sanskrit it is called 'sam-pra-san-na'.
'abel' would mean something that does not 'cling' or has 'affinity'.
The Hindi words for the roller-pin, "belan(a)" and for the vine or the creeper, "bel", are also connected with the root.