Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra

Let an algebra P over the field F of characteristic not two be any Lie-admissible algebra, permitting a non-degenerate composition with symmetric bi-linear trace (or associative) form. Then, defining a new deformed product and a new symmetric bi-linear form in P, the deformed algebra P is shown to be alternative with the unit element. Moreover, it permits non-degenerate composition again now with respect to the new deformed product. Hence, by the Hurwitz theorem, only possible dimensions of the original algebra P are limited to 1,2,4, and 8. Further, the pseudo-octonion algebra P/sub 8/, the pseudo-quaternion algebra P/sub 4/, and the pseudo-quadratic algebra P/sub 2/ transform into the usual octonion, quaternion and quadratic algebra, respectively, by means of the deformation.