aren't both the rear wheels "Bolt-on" style with 3 bolts.... ???Yes and No. you have an open differential which allows one wheel to slow down when making turns without scuffing the grass. With the key missing the wheel with the least resistance in turning will receive all the power and in your case will cause the axle to spin inside of the rim without the key transferring no power to the other wheel.
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I am thinking it is this model not the LT-1000 you listed https://www.manualslib.com/manual/496554/Craftsman-917-272460.html?page=38#manualaren't both the rear wheels "Bolt-on" style with 3 bolts.... ???
Craftsman LT 1000 (LT 1000-14) - Craftsman Lawn Tractor (1991-03) Engine Mount Assembly Parts Lookup and Diagrams | PartsTree
Find genuine OEM replacement parts and diagrams for the Engine Mount Assembly for the Craftsman LT 1000 (LT 1000-14) - Craftsman Lawn Tractor (1991-03)www.partstree.com
Yes, there's a square key that you need. Because of the differential, a free spinning wheel will prevent movement of the other wheel. I have LT1000 myself.I have a LTX1000 Craftsman mower. The right rear wheel fell off. When I reinstalled the mower no longer engages. I understand there is a square key that probably fell out. If that's the problem is the mower only driven by the right rear wheel?