On the construction of reductive Lie-admissible algebras

AbstractWe discuss a method to construct reductive Lie-admissible algebras which is based on the construction of nonassociative algebras with a specified simple Lie algebra D of derivations. As special cases, we construct two classes of reductive Lie-admissible algebras (A,∗) of dimensions 7 and 8 with D=sl(3) and D=G2, and determine their associated reductive Lie algebras g−=A−⊕sl(3) and g−=A−⊕G2. The split octonion, para-octonion, 7-dimensional simple Malcev algebra and simple Lie algebras of type A3, G2, B3 arise from this construction. Representations of simple Lie algebras play a main role