# Structurable algebras of skew-dimension one and hermitian cubic norm structures

## TL;DR

This paper explores structurable algebras of skew-dimension one, providing new constructions and explicit formulas, and connecting these algebras to known processes like Cayley-Dickson, with some results previously unnoticed.

## Contribution

It introduces two equivalent constructions for these algebras, offers explicit norm formulas, and links them to the Cayley-Dickson process, expanding understanding of their structure.

## Key findings

- Every form of a matrix structurable algebra can be described by the new constructions.
- Explicit formulas for the norm $
u$ are provided.
- A precise connection with the Cayley-Dickson process is established.

## Abstract

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the norm $\nu$; (3) We make a precise connection with the Cayley-Dickson process for structurable algebras.