[This is an expanded re-posting of a question I asked in April 2020, that subsequently got auto-deleted as an "abandoned question." Hopefully launching a second attempt now (over two years after I first asked it, and I believe over a year after it was deleted) is okay, particularly since I've expanded it with variables, equations, etc. to "flesh out" or clarify what I had written earlier.]
I have a BA in Mathematics from the University of Maine at Farmington, but am in no means a professional mathematician and my knowledge of things like tessarines (bicomplex numbers) and split-complex numbers is from Wikipedia. This question is kind of a question of mathematical terminology when dealing with division algebras that are (in a certain sense) "products" of other division algebras. I know what it means when a group is a direct product of two groups of lower order, but when it comes to (hyper)complex numbers I'm not sure of the jargon.
Tessarines are commonly written in the form $w+xi+yj+zk$ (with possible different coefficients than $w$, $x$, $y$ and $z$ or different order of those coefficients, but I'll use these coefficients here), where $ij=ji=k$, $i^2=-1$, and $j^2=+1$. So you have a real number component (possibly $0$), a real coefficient (possibly $0$) of the same $i$ used in the complex numbers, a real coefficient (possibly $0$) of the same $j$ used in the split-complex numbers and a real coefficient (possibly $0$) of $k$ which is the product of $i$ and $j$.
Introducing complex variables $α=w+xi$ and $β=y+zi$ and split-complex variables $γ=w+yj$ and $λ=x+zj$, tessarines can be written in the forms $γ+λi$ or $α+βj$. So, if you substitute split-complex number coefficients for real coefficients in the common form of the complex numbers, or if you substitute complex number coefficients for real coefficients in the common form of the split-complex numbers, you get the tessarines, or at least an algebra isomorphic to the tessarines which is fairly easily converted into the common tessarine form I described above.
So, what does that make the relationship between tessarines, complex numbers and split-complex numbers? The tessarines are the [blank] of the complex numbers and the split-complex numbers. What's the [blank]?
I know tessarines are equivalent to bicomplex numbers, which are sometimes written using the basis {$1$, $h$, $i$, $hi$}, where $i$ can be thought of as the same $i$ used above (it squares to $-1$), $h$ (which also squares to $-1$ and commutes with $i$) is equivalent to $k$ and $hi$ is equivalent to $-j$. These bicomplex numbers have some kind of relationship between the complex numbers and (the complex numbers) themselves. What's that relationship called? The bicomplex numbers are the [blank] of the complex numbers and (the complex numbers) themselves? What's that [blank]?
I think the answer to one of these two questions may be the tensor product, but I'm not sure which one. Wikipedia's article on hypercomplex numbers ( https://en.wikipedia.org/wiki/Hypercomplex_number#Tensor_products ) states that bicomplex numbers (and thus tessarines) are the tensor product of the complex numbers and (the complex numbers) themselves, but I would think of tessarines as more of a product of the complex numbers and the split-complex numbers. But I know algebraic structures often have multiple types of "products."
Thanks to anyone who can help in answering my question here.
Best Answer
The tensor product (of algebras), over $\mathbb{R}$.
It's still the tensor product again. This may be a little surprising; it shows that the tensor product isn't cancellative.