I found this thread when I was trying to replace the cartridges in my Brizo Levior bathroom faucets. Include me into the fact that there is a spline adapter that fits around the cartridge in these Brizo handles. Desplaine adapter looks like a thick gear with a fine grooved circumference at the base. there is a screw at the very top that attaches the spline adapter to the top neck of the cartridge.
In my particular case, the spine adapter model number is RP 92382. Which connects to cartridges RP100222 (hot) and RP100223 (cold).
When removing one of the handles, the entire spline adapter seems to have bonded with the interior surfaces of the handle, making the removal far more difficult than just pulling up and breaking the magnetic seal. Maybe it was glued by an unknown plumber before me. I don’t know. It sure made my afternoon difficult to figure it all out.