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A few days ago, I post a question about the derivatives of these units. But to do it, I need to know everything about base units. I think this forum is optimal to collect our knowledges of it.
So a multidimensional number is from a linear combination of these units,
We multiply them like this way: 
where i!=j,  is the levi-civita tensor which gives the sign of the multiplication, and ^ is a logic binary operator: xor.
And of course, we can represent them as matrices, too ... But just from reals to quaternions, because of the non-associativity of the octonions, the multiplication of the matrices of the representation of these octonions does not work ... I do not know why. (?)
I have many questions:
Why cannot we represent the octonions, sedenions ... etc.?
What is the derivative of ?
What is (inverse)?
Which logic binary operators could we substitute instead of ?
What is the taylor series of the xor and the levi-civita ops?
How can we calculate with i[x] where x is not integer, so it is real or complex?