British Columbia: Transplanting/Dividing a 'Black Beauty' Lily

My Black Beauty(Oriental x Trumpet) Lily was bought as a single bulb about 20 years ago. It has done well and is now a large clump with about 8 tall stems close together, plus a single sturdy a stem which regularly emerges from the soil about 2 feet away from the others. (see photos) All the stems flowered profusely this year.

By its size and location I assumed this stem was coming from a separate bulb and was planning to dig it up and give it to a friend. Knowing that, (unlike tulips) lily stems often don't grow up from the top of the bulb I started a careful excavation to avoid damaging the bulb, which I thought might be a few inches away from the stem. So far my digging has exposed about 14" of underground stem which grows down for about 6" and then turns and runs almost horizontally towards the main clump. Does this mean that there is no separate bulb/plant that can be transplanted? Or if I just keep following the underground stem, will I find a bulb I can give away? Will I have to cut it away from the main plant, and if so how do I avoid damaging the main clump? Is it OK to do the division/transplant at this time of year in Vancouver?

If it can't be separated and given away, can I do something to make this stem come up close to the main clump?

 

You will encounter a bulb at the end of it at some point. And lily stems do come off the top of the bulb.

 

Thanks, It is good to know that each stem represents a separate, and separable bulb.
Regarding the stem I didn't mean to say that they didn't come off the top, but rather that (in my experience) lily stems often seem to bend sideways underground and emerge from the soil off to one side, so that you have to be careful when digging them up not to assume that the bulb is directly below the above-ground stem.