  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.
