The referred-to book can be previewed on Google Books. Searching the book finds the following (unsourced) claim:

In the eighth century however, recognizing the sanitary value of linen garments, Charlemagne revived the flax industry.

The author on that page goes into some detail about how linen is actually produced from flax (TIL).

No mention in the book of Charlemagne's recognizing the health benefits of flax seed. Flax seed oil is not mentioned in my medieval food reference books in connection with Charlemagne (nor much at all); I only find references to production of linen cloth in my textile and many economic references of this period. I don't know all of Charlemagne's capitularies (edicts), but I think this health claim is a case of willful ignorance of the actual historical record. Moreover, Charlemagne was not in the habit of establishing laws around food outside of certain Christian requirements and certain decisions around 'planning the economy'.