To add on to your mention of the rotation in abstract space , this is a local transformation of the electromagnetic potential. Not saying that "rotation" is a terrible thing to call it. it's just not usually thought of as a literal rotation. How about "twisting the potential"? Eg "twist" electric field into magnetic field? Rotation would connote that this is not 1D.

Some also think of this additional Lie as a ("central") extension of the Galilei group?

https://physics.stackexchange.com/questions/281485/how-did-m...

(Sorry, couldnot get Gemini to give a ref for that)

Update: better ref, but paywalled

https://pubs.aip.org/aapt/ajp/article-abstract/48/1/5/235124...