Real Division Algebras with a Left Unit Element that Satisfy Certain Identities

We study A, finite dimensional real division algebra with left unit e , satisfying: for all x E A , (E1) E1 ) (x, x, x, x ) = 0, (E2) E2 ) ( x 2 , x2, 2 , x 2 ) = 0, (E3) E3 ) x2e 2 e = x2 2 and (E4) E4 ) (xe)e xe ) e = x . We show that: center dot If A satisfies to (E1), E1 ), then e is the unit element of A . center dot (E1) E1 ) =double right arrow double right arrow (E2) E2 ) =double right arrow double right arrow (E3) E3 ) =double right arrow double right arrow (E4). E4 ). In two-dimensional, we determine A satisfying ( Ei ) i is an element of{ 1 , 2 , 3 , 4 } . We have We show as well as (E1) E1 ) =double right arrow double right arrow (E2) E2 ) double left arrow double right arrow (E3) E3 ) =double right arrow double right arrow (E4). E4 ). We finally study the fused four-dimensional real division algebras satisfying ( Ei ) i is an element of{ 1 , 2 } . We have shown that those which verify (E2) E2 ) are H , * H and C B . and that H is the only fused algebra division with left unit satisfies to (E1). E1 ).