Have used flaxseed (not chia seed) gel in my soaps for years.
I think it is moisturizing. It keeps the batter gel like in HP which makes it so much easier to pour into cylindrical molds.
If using the seeds, best to cook seeds as they can get rancid over time. Haven't made a batch since I started using citric acid and wonder if the citric acid will reduce the rancidity (as it can with linoleic acid in oils).
Otherwise, the gel alone has worked fine most of the time. ... Saw some strange things in 2 batches. One of over 1.5 years, saw one or two small brown marks. Always wondered if the flaxseed gel was responsible. That batch had avocado puree, shea butter among other fats with a SF of 10% ... along with the flaxseed gel The other was the batch I saw mold in and figured the problem was the carbohydrates (oats and sugar) I also included. Unsure whether flaxseed is food for mold because the much older avocado soap never got mold, though I did treat that avocado batch better as I was not trying to acid test it re bad consumer practices. Happy for thoughts.
All in all and despite those 2 unfortunate cases, I love flaxseed in soap.