I believe its also in the water supply in certain places, so if it works for dementia there are natural experiments already running on this.
https://www.psychiatrictimes.com/view/low-dose-lithium-a-new...
https://www.cambridge.org/core/journals/the-british-journal-...
I would think naturally occurring lithium in some people's water would give pretty good control conditions to do a wide study of this effect on Alzheimers as well?
These things are simple enough to advise the populace to use on their own. The government should never play nanny, ever.
Results
The literature search identified 415 articles; of these, 15 ecological studies were included in the synthesis. The random-effects meta-analysis showed a consistent protective (or inverse) association between lithium levels/concentration in publicly available drinking water and total (pooled β = −0.27, 95% CI −0.47 to −0.08; P = 0.006, I2 = 83.3%), male (pooled β = −0.26, 95% CI −0.56 to 0.03; P = 0.08, I2 = 91.9%) and female (pooled β = −0.13, 95% CI −0.24 to −0.02; P = 0.03, I2 = 28.5%) suicide mortality rates. A similar protective association was observed in the six studies included in the narrative synthesis, and subgroup meta-analyses based on the higher/lower suicide mortality rates and lithium levels/concentration.