Lily bulb roots
New roots are grown each year from the bottom of the bulb form the basal plate, and along the underground stem portion. Basal plate roots go deeply into the ground to help anchor bulbs against the effects of high wind, for accessing deeply buried nutrients and ground moisture, and for your plant's long-term health. This "contractile" root system actually pulls bulbs deeper in light soil. When dividing your bulbs, many bottom roots are lost, which are not as important as roots formed above the lily bulb - stem roots.
Encouraging the formation of stem roots after transplanting should be your most important, immediate goal. These critical roots need nutrients within the top 2 or 3 inches of soil - where a top-dressing of fertilizer, compost or well-rotted manure can be placed, and where nature provides nutrients in the wild. If lilies are not planted deep enough these roots may either not form or be exposed to air, and will not benefit the lily.