From what I gather. Wood ash produces Calcium Hydroxide or Potash. Commercial "Lye" is Sodium Hydroxide. You can't get commercial lye from wood ash. You get it from Calcium Carbonate and other chemicals. Sorry but your chemistry is sooo wrong. If you take wood ash and leach it you get POTASH which is a mixture of various salts mostly based on POTassium. Sodium is not used buy most woody plants but those that will grow with low amounts of sea (salt) water should contain more and produce more sodium hydroxide in the ash and potash. Sodium Hydroxide today is produced from sodium chloride solution by passing a DC current through the solution. Chlorine is driven off as gas and the sodium attaches itself to the water. Calcium Hydroxide is often called LIME a substance with many uses but I don't think soap making is one of them. In the old days Lime was produced by burning bones, shells or rocks that were rich in Calcium [limestone] this would produce calcium oxide and when mixed with water would go to calcinm hydroxide ~gd
Calcium Hydroxide produces a softer soap which now days most people use for liquid soap. I'm not sure but I think you can add salt to make the soap harder. We use to do that in the shampoo factory to make a thicker product. Might work with a bar of soap I'm not sure. Also you could add more fats like Lard and such to make the soap more solid.
I tried to make soap years ago but I could never find pure Sodium Hydroxide easily. So I kind of gave up on the whole hobby. I figure that I could make a decent bar of soap today but it's much easier to get it from the store. I'm lazy like that.