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For example, normally one just uses, the coordinates (x,y,z,it) with i^2=-1. This works fine with one time coordinate. Using two corrdinates one may suggest (x,y,z,it,jw) where j^2=-1 too. Using iw would make the time coordinated indistinguishable. Now there is an algebraic problem; what is i*j? Hamilton (mid 1800s) discovered that there must be another term (k) where k^2=-1 and to be consistent, ij=k.

Why did I read this far? I don't know. My eyes had already glazed over.

If i and j are equal to the same thing (sqrt -1), why are they not equal to each other? Why / how can ij = k (which is also sqrt -1) if ij = i^2 = -1?

Am I misreading the * operator?

It's the kind of thing that happens in vectors. In ordinary (Gibbs) vector analysis, the three spatial coordinates get unit vectors i,j,k attached. Hamilton used 4 coordinates for the quaternions. (There isn't a 3-dimensional division algebra.) with a basis (1,i,j,k). One gets that i^2=j^2=k^2=-1 but ij=k. Using this rule, the product of two quaternions (0,iy,jz,kw)*(0,ib,jc,kd) would have the first term as the negative of the dot product and the last three terms as the cross product as if the (y,z,w) and (b,c,d) were ordinary 3-vectors. It's kind of weird. If one trys to expand the complex numbers to 3-space analogously to expanding the reals to complex, then one finds that 4-coordinates are needed. (Algebra is full of surprises.)

I'm just speculating here. If one has two time-like dimensions, it seems that one then really needs three such.