In the sixteenth century, the Gaelic people of Ireland and the Highlands of Scotland wore a style of short jaket called an ionar. This jacket had sleeves that were open along the bottom, and either tied or buttoned at the cuff.

The reason for this was because the shirts they wore at the time (called in Gaelic leine), had extremely long, full sleeves.
http://albanach.org/leine.html

You can see pictures in the article linked above to men wearing the ionar with the leine. Sometimes the sleeves are worn loose, sometimes fastened at the wrist.

I suspect jackets that are today made with split sleeves are modeled after the sixteenth century ionar.

Aye,
Matt