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)_i∈\1,2,3,4\. We havetabular|c|c|c|c|c| % after \\: or col1-col2 col3-col4 ... A \ satisfies to & (E1) & (E2) & (E3) & (E4) \\ A \ isomorphic \ to &R$; $C$ & $R$; $C$; $^{}C$ & $R$; $C$; $^{}C$ & $R$; $C$; $^{}C$; $L(1, -1, γ, 1)\\ tabular We showas well as(E1) (E2) (E3) (E4).\ finally study the fused four-dimensional real division algebras satisfying (Ei)_i∈\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).