elements in different spirits

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Dear Mr Stroud,
I have not been able to read the Golem's eye or the ring of Solomon, though I've read the other two. So, I don't know if this is addressed.
As per your definition, elementals, are one element spirits, while remaining spirits, (who I always think of as daimons, being a politer term than demons) normally have two elements of greater quantity. I have made some simple calculations, and, the 5 middle ranks of daimons are not accounted for.
Here are the possible combinations:
(Key: F=fire, W=water, E=earth, A=air)
FE, FA, FW, WE, WA, EA which is a total of six. Now we know, from the discussions on mixed elemental spheres, that every element has an opposite, thus no of broad categories of possible daimons=6-2=4! , which does not account for imps, foliots, djinn, afrits & marids. So are there 3 element combinations as well?

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