Just prepare a small pot for the 'baby', with lots of grit in the soil for fast drainage (more grit than soil) and using a sharp knife, cut right between the mother and baby, making sure the baby has some roots (you'll have to dig carefully to 'allot' some before cutting) and plant it.

Why divide? I went to a private garden (but open to the public) in Miami, FL back in the mid 80's. This garden, I believe, has the largest US colection of palm trees. At the entrance was a ponytail palm that covered an area about 15' x 15'. The caudex (water holding part of the root)wasn't more than a foot above the ground level. There were several 'ponytails' rising from the clump of caudex. It was an impressive site! Your bonzai ponytail will not grow beyond it's container, when trimmed properly, and would make an impressive display. There are others who know more about bonzais than I do, and may make other recommendations. Enjoy your ponytails!

These two palms each grow one palm from one seed they do not have any new palms growing from there roots.You could have another palm all together that is growing there. Can you show a picture ? About the names Alexander was a common name given to a archotophoenix alexandrae and some time or another people started to call Ptychosperma elegans the same common name. The other common name that is more familiar for Ptychosperma elegans is Solitaire Palm . Christmas palm is a common name for Adonidia merrillii which was once thought to be a veitchia. Adonidia merrillii looks very similar to Ptychosperma elegans when young. Archontophoenix alexandrae are sometimes planted with multiple seeds in the pot and some trunks will grow fast and others very slow so it gives the appearance the are(multipliers)but they are not. But if you can post a pic we can clear that up too if you like.Here is a picture of one of mine.Its the big palm on the left. And this is a Ptychosperma palm (not elegans) but similar in appearance and it does sucker or multiply..

Those are individual plants. Palm growers plant several seedlings in a pot to get a clump like that. I can only tell you how I've seen it done at nurseries I've worked at. The pot will be pretty rootbound. If there is a mass of roots around the sides and you can't "tease" the plants apart, just slice about 1/4 inch off the sides and about 1/2 inch off the bottom to free up the root ball. Then bang the root ball on the ground hard until the root ball loosens up. Then try to tease the plants apart. You can use a turning fork or pitchfork to work in between the plants and separate them. You're going to damage some roots, but that's o.k. Once you get them separated (this may take awhile, so be patient), cut off all but the top three fronds, in other words, "rooster tail" each plant. Prune the bottom half off of the largest bottom frond. Put them into individual pots or where you want them to be, and water well. If you just want to make two separate clumps, slice the ball in two, tease the bottom roots out and plant each clump individually.