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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 analgebras. 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 algebrasnot yet specified in the literature.

Classification of the Absolute-Valued Algebras with Left-Unit Satisfying x2(x2)2 = (x2)2x2