Then again maybe dimension 4 is too small and simple to be interesting.
It does not even have a square root if we use real coefficients.
( they have 8 solutions over the complex coefficients though )
Now it turns out commutative nonassociative hypercomplex numbers that have a square root must be of dimension 4n.
Powerassociative is also a thing.
Im not an expert in this though.
Informally speaking , the main issue is constructing such cayley tables such that it does not have any " subcayleys " (the analogue of subgroup), yet do have a structure that makes the square root possible.
In dimension 8 we have for instance this
(see picture)
But Im not sure about the properties yet.
8 = 4*2 , so the complex might be a subgroup and then the other subgroup would be one of those 4 dim ones.
( edit : it has no subalgebra , this has been checked see follow up replies )
sqrt has not been investigated.
I think it is power-associative.
regards
tommy1729
It does not even have a square root if we use real coefficients.
( they have 8 solutions over the complex coefficients though )
Now it turns out commutative nonassociative hypercomplex numbers that have a square root must be of dimension 4n.
Powerassociative is also a thing.
Im not an expert in this though.
Informally speaking , the main issue is constructing such cayley tables such that it does not have any " subcayleys " (the analogue of subgroup), yet do have a structure that makes the square root possible.
In dimension 8 we have for instance this
(see picture)
But Im not sure about the properties yet.
8 = 4*2 , so the complex might be a subgroup and then the other subgroup would be one of those 4 dim ones.
( edit : it has no subalgebra , this has been checked see follow up replies )
sqrt has not been investigated.
I think it is power-associative.
regards
tommy1729