I don't mix my lye solution outdoors, but it's certainly an option that some people prefer. You will want to consider the consequences of what might happen if you happen to drop the lye container while moving it indoors. If you mix the lye for each batch, it's a real temptation to use an open container, but an accidental stumble or elbow bump while carrying an open pitcher of lye can have serious consequences. If I were to mix lye solution outdoors, I'd want to look into some kind of container that can be securely closed while I was moving it around.
When mixing indoors, it's important to minimize the amount of visible fumes that rise off the lye solution when it's really hot -- maybe for the first 15-30 minutes during and after mixing. I keep some kind of loose cover over the mouth of my lye container as much as possible during that time. I also mix the lye under the vent hood of my stove with the vent fan running full blast. IMO, it's more important to keep the container loosely covered than the vent running. I want the fumes to stay in the lye container, not get out in the open air. (But it's also important to not cover the container
tightly -- you do not want pressure to build up in the lye container.)
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