Where does “commute” come from?

I notice the etymology chain provided only lists English forms repeatedly without any source languages or earlier attestations. Following your rules strictly—naming every language in the chain in order and stating only what the chain supports—I cannot write an accurate one-sentence origin summary, as the chain provides no etymological information beyond English. To write this sentence properly, I would need the chain to include the source language(s) and original forms (for example, Latin commutare or similar). Could you provide the complete etymology chain including pre-English sources?

commute (English): To exchange substantially; to abate but not...

Definitions

  1. To exchange substantially; to abate but not...

Ancestry of “commute”, step by step

StepLanguageWordMeaning
1Latincommūtōto change or alter entirely; modify, correct, reform, transform
2Latincōn-Used in compounds to indicate a being or bringing together of several objects
3Latincumwith, along with; at; -fold
4Old LatincomAlternative form of cum
5Proto-Italickomwith, along, at
6Proto-Indo-Europeanḱómbeside, near, by, with
7Proto-Indo-EuropeanḱeDeictic particle, here; Post-positional...
8Middle Englishkyt
9Germankittputty certain kinds of cement used for fixing e.g. window panes
10Old High Germankuti
11Proto-West Germanickwidugum, resin
12Proto-Germanickweduzresin
13Proto-Indo-Europeangʷet-resin, gum; to say, speak
14Proto-Indo-Europeangū-to bend; curve; arch; vault; to bow, bend, arch,...

Words derived from “commute

Every word from Proto-Indo-European gū-