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We study A, finite dimensional real division algebra with left unit e, satisfying: for all xA, (E1) (x,x,x)=0, (E2) (x^2,x^2,x^2)=0, (E3) x^2e=x^2 and (E4) (xe)e=x. We show that: If A satisfies to (E1), then e is the unit element of A. (E1)(E2)(E3)(E4). In two-dimensional, we determine A satisfying (Ei)_i{1,2,3,4 . We have tabular|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 show as well as (E1)(E2)(E3)(E4). We finally study the fused four-dimensional real division algebras satisfying (Ei)_i{1,2 . We have shown that those which verify (E2) are H, ^H and CB. and that H is the only fused algebra division with left unit satisfies to (E1). , number=3, journal=European Journal of Pure and Applied Mathematics, year=2024, month=Jul., pages=2276–2287