Classification of the Absolute-Valued Algebras with Left-Unit Satisfying   x2(x2)2 = (x2)2x2
  We show that every absolute-valued algebra with left-unit satisfying (x2; x2; x2) = 0 is finite-dimensional of degree at most 4: Next, we determine such an algebras. In addition to the already known algebras R; C; \astC; H; \astH; \astH(i; 1); O; \astO; \astO(i; 1) the list is completed by two new algebras not yet specified in the literature. 
Alassane Diouf, Maribel Ramirez and Abdellatif Rochdi
math.RA