Nonassociative algebras of biderivation-type

The main purpose of this paper is to study the class of Lie-admissible algebras $(A,.)$ such that its product is a biderivation of the Lie algebra $(A,[,\left]\right)$, where $[,]$ is the commutator of the algebra $(A,.)$. First, we provide characterizations of algebras in this class. Furthermore, we show that this class of nonassociative algebras includes Lie algebras, symmetric Leibniz algebras, Lie-admissible left (or right) Leibniz algebras, Milnor algebras, and LR-algebras. Then, we establish results on the structure of these algebras in the case that the underlying Lie algebras are perfect (in particular, semisimple Lie algebras). In addition, we then study flexible $A_{BD}$-algebras, showing in particular that these algebras are extensions of Lie algebras in the category of flexible $A_{BD}$-algebras. Finally, we study left-symmetric $A_{BD}$-algebras, in particular we are interested in flat pseudo-Euclidean Lie algebras where the associated Levi-Civita products define $A_{BD}$-algebras on the underlying vector spaces of these Lie algebras. In addition, we obtain an inductive description of all these Lie algebras and their Levi-Civita products (in particular, for all signatures in the case of real Lie algebras).