If it is a Cattleya hybrid as you suspect (posting a picture would tell us for sure) it will probably not re-root. What it may do is start a new growth and that new growth will root. A 1-bulb division is a small division, the recommended viable division size is usually 3+, so you will probably have a bit of a wait on your hands before you get any flowers.
Here's what I'd do with it:
1) Get a small pot, probably 2-inch is big enough and fill it iwth standard orchid bark mix.
2) Secure a stake in the middle of the pot.
3) Place the single pseudobulb, with the base setting firmly on the mix, and secure it to the stake using twist-ties or string.
4) Wait for a new shoot to appear at the bottom. When the new shoot starts to grow roots, begin watering. There's no point in watering until the plant has roots because theplant has no way of absorbing any moisture.
Get a good Cattleya culture sheet like the one found at
AOS | Home. For now, give it slightly less light than the clture sheet recommends.