Here is a Goggle link that has several photos of suffruticosa if you're concerned about the name, these photos might help. http://images.google.com/images?q=sansevieria+suffruticosa&ie=UTF-8&hl=en&btnG=Google+Search
Or you can just do an image search on Google like I did.

As for cutting the rhizomes. That's a personal choice. Personally, I like the looks of the rhizomes and possibly allowing the rhizomes to grow might encourage the plant to flower.

Elsie, I found a long narrow pot, and tried to get them to root in the same pot as the parent... They progressed only so far, and well, I think it was a dismal failure. BUT, that could be because I was pretty much ignoring the plant, and it really looks it these days. I have it out, front and center now, so it should get better water and more attention.

I suspect it doesn't matter. Plants will follow their biological imperatives, whether we interfere or not. Do what ever you like the look of.