Absolute valued algebras satisfying (x,x,x2) = 0

Let A be an absolute valued algebra satisfying the identity (x,x,x2) = 0. We give some conditions which imply that A is isomorphic to R, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \mathbb{C} $\end{document}, H or D. These results enable us to show that if A is an algebra with involution then A is one of those classical algebras. We construct an example of A having dimension two and is not isomorphic to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \mathbb{C} $\end{document}.