Does flax seed lose its nutritional properties after it's ground up? I am from Santiago, Chile. Here, there's a rumour that smashed (ground) flax seed loses its properties in around 4 hours. Do you know anything about this?

Basically, the rumors say that the flax seed loses its nutritional properties, and that it can even be unhealthy once it is ground — but, if you grind it yourself and eat it right away, it's OK.
Any seed that has been damaged, cut, smashed, milled, or ground starts to lose flavour, texture, nutrition, and eventually will go rancid due to oxidising oils.

Four hours is too short a time for anything noticeable to happen. Some types of nuts and seeds show a noticeable change over a day or two, but most take many days or weeks.

Milled flax seed is reasonably tough; it should last for a week or so without a noticeable flavour change, and many weeks before going rancid. For more information and sources, please see: https://www.goldenvalleyflax.com/flax-facts/health-research-articles/flaxseed-storage-baking-stability/