### No New Normed Division Algebra Found!

#### Posted by John Baez

Good news! The paper mentioned in my last article here, Eight-dimensional octonion-like but associative normed division algebra, has been retracted:

- Scott Chapman, Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra,
*Communications in Algebra*, 19 October 2020.

Here’s the retraction notice:

We, the Editors and Publisher of the journal Communications in Algebra, have retracted the following article:

“Eight-dimensional octonion-like but associative normed division algebra” (DOI: 10.1080/00927872.2020.1791899).

Since publication, concerns have been raised about the integrity of the mathematics in the article. The main error is visible not only in the title, but in the abstract which claims “We present an eight-dimensional even sub-algebra of the $2^4$ = 16-dimensional associative Clifford algebra $\mathrm{Cl}_{4,0}$ and show that its eight-dimensional elements denoted as $X$ and $Y$ respect the norm relation $\|X Y\| = \|X\| \|Y\|$, thus forming an octonion-like but associative normed division algebra.” In short, the author claims to have found an 8-dimensional normed division algebra over $\mathbb{R}$, but there is no such thing. Hurwitz’s theorem says all 8-dimensional normed division algebras over the reals are nonassociative and isomorphic to the octonions. This famous result, published in 1923, has been confirmed with a number of well-established proofs. Thus, based on these factors the Editors assess the author’s main result to be false.

The Editors have discussed this with the author, and sought additional review of these points of concern. As the main result cannot be relied upon, the Editor and Publisher are retracting the article. The author has been notified of the retraction and does not agree with this decision. We have been informed in our decision-making by our policy on publishing ethics and integrity and the COPE guidelines on retractions.

The retracted article will remain online to maintain the scholarly record, but it will be digitally watermarked on each page as “Retracted”.