Important correction: A division algebra is an algebra in which every non-zero element has an inverse. The dual numbers for instance are a Geometric Algebra which are not a division algebra because there are some non-zero dual numbers which don't have inverses. In fact, almost all Geometric Algebras fail to be division algebras.*
So your point about division algebras is not particularly relevant to the article.
* - Frobenius's theorem classifies all the finite-dimensional associative division algebras. They are: The real numbers, the complex numbers, and the quaternions. There are no others.
So your point about division algebras is not particularly relevant to the article.
* - Frobenius's theorem classifies all the finite-dimensional associative division algebras. They are: The real numbers, the complex numbers, and the quaternions. There are no others.