You could, but then you would be at a loss as how to attach a sprocket to the bottom bracket spindle.
The B-E.com sprocket uses a cotter key and nut to hold the sprocket onto the spindle.
**Lame.**
The SBP spindle has a keyway machined into it to accept a woodruff key and their sprocket adapter also has a matching keyway ( and the woodruff key comes with it).
This is a very strong and reliable method of attachment.
You could use the B-E.com cheesy sprocket on the SBP spindle if you filed down a flat spot on the spindle to accept the cotter key. I highly recommend against this though.
Cotter keys are very old technology that work well...once. Take them apart and most of the time you have problems re-assembling them later.
Take my old setup for example, the cotter key received a deep gouge in it and at the same time rounded the edge off of the matching flat spot on the cheesy soft B-E.com spindle.