If you look around I think you can find some SJ or SH spec oils in the implement stores and maybe in the lawnmower section. The Rotella T-4 oil seems to be fine for about any engine that doesn't call for the latest specs. But watch for "meets or exceeds" on anything you buy. Might wind up with a cylinder bore like this:
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Might notice the wear is below the rings, piston itself is scoring the cylinder walls. Poor lubrication on the Kool Bore Briggs.