Keep in mind that there is a specially formulated oil for air cooled engines such as mowers, tractors, some motorcycles and yes, splitters.
I don't find any special oil requirements other than what is below. SG (SF) is a dirt common rating.
Virtually all small kohler engines require SF (Old) or SG rated oils with a viscosity appropriate for the temps it is operated in.
SG – 1989 Gasoline Engine Warranty Maintenance Service –
Service typical of gasoline engines in cars, vans and light trucks, beginning
with the 1989 model year operating under manufacturers’ recommended maintenance procedures. Provides improved control of engine deposits, oil oxidation, sludging and engine wear relative to previous categories. Rust and bearing corrosion limits are the same as SF oils. May be used where SF, SE, SF/CC or SE/CC is recommended. All of LE’s MONOLEC Engine Oils exceed this classification.
(Exert stripped from here: (broken link removed to http://www.motorexbih.com/API-SAE%20Engine%20Oil%20Service.pdf))
There really isn't anything special about the oil that I can find. Open ears - willing to learn.
Just to check myself I reviewed service manuals for three different familys of Kohler engines and they all had pretty much the same language for the oil:
Other high-quality detergent oils (including synthetic)
of API (American Petroleum Institute) service class SJ
or higher are acceptable. Select viscosity based on
air temperature at time of operation as shown in table
below.
(stripped from manual here: (broken link removed))