By means of principal isotopes l H(a, b) of algebra l H [Ra 99] we give an exhaustive description of all 4-dimensional absolute valued algebras satisfying (x p , x q , x r ) = 0 for fixed integers p, q, r ∈ {1, 2}. For such an algebras the number N (p, q, r) of isomorphisms classes is either finite included between two and three, or is infinite. ConcretelyBesides, each one of above algebras contains 2-dimensional subalgebras. However, the problem in dimension 8 is far from being completely solved. In fact, there are 8-dimensional absolute-valued algebras, containing no 4-dimensional subalgebras, satisfying (x 2 , x, x 2 ) = (x 2 , x 2 , x 2 ) = 0.Keywords. Absolute valued algebra, central (flexible) idempotent, principal isotopes of l H. * l H . In our computations the first purely imaginary element i in the canonical basis {1, i, j, k} of algebra l H, will be used.
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