Real Division Algebras with Left Unit Satisfying Some Identities.

We study A, finite dimensional real division algebra with left unit e, satisfying: for all x A, \\\ (E1) \ \ (x, x, x) =0, \ \ \ (E2) \ \ (x², x², x²) =0, \ \ \ (E3) \ \ x²e=x² \ \ and \ \ (E4) \ \ (xe) e=x. \ show that: If A satisfies to (E1), then e is the unit element of A. (E1) (E2) (E3) (E4). \ two-dimensional, we determine A satisfying (Ei) ₈\₁, ₂, ₃, ₄\. We havetabular|c|c|c|c|c| % after \\: or col1-col2 col3-col4. . . A \ satisfies to C C; ^C C; ^C C; ^C; L (1, -1, , 1) \\ tabular We showas well as (E1) (E2) (E3) (E4). \ finally study the fused four-dimensional real division algebras satisfying (Ei) ₈\₁, ₂\. We have shown thatthose which verify (E2) are H, ^H and C B. and that H is the only fused algebra division with left unit satisfies to (E1).