On a problem by Nathan Jacobson

Abstract

We prove a coordinatization theorem for unital alternative algebras containing $2\\times 2$ matrix algebra with the same identity element 1. This solves an old problem announced by Nathan Jacobson on the description of alternative algebras containing a generalized quaternion algebra $\\mathbb{H}$ with the same 1, for the case when the algebra $\\mathbb{H}$ is split. In particular, this is the case when the basic field is finite or algebraically closed.