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I am currently considering taking Iosol (ammonium iodide and Iodine/I2) or Lugol's iodine (potassium iodide and Iodine/I2).

Lugol's solution: formulated in 1829
Iosol: formulated in 1945

Solubility of ammonium iodide in water at 20 degrees Celsius: 172g/100g
Solubility of potassium iodide in water at 20 degrees Celsius: 140-148g/100g

According to this article https://iodinehealth.wordpress.com/iodine-types/ Iosol was formulated with ammonium iodide because according to studies of Hashimoto's thyroiditis in Japan among people who consume too much seafood (which contains potassium iodide) potassium iodide can cause thyroid congestion leading to Hashimoto's, but the article says these studies were disproven.
In the comments of the article if you ctrl+f "March 17, 2011 2:26 pm" you can find a comment which says that the body converts all forms of iodine into potassium iodide before it can be utilized by the thyroid. Is this true? If it is, then Lugol's solution would be better than Iosol.

According to this article https://www.supporthealth.com.au/health/iodine/ excess ammonium is cleared away as part of the urea cycle similar to many other body processes. Because of this and the relatively high solubility in water compared to potassium iodide, I would think Iosol/ammonium iodide would be a better choice.

Something else to consider is that Iosol uses glycerin as a solvent whereas Lugol's just uses water.