We introduce the notions of anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, which arise as nearly associative Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras, respectively. We establish a correspondence between these classes of algebras and the Levi-Civita products of their associated Lie and Jordan structures.
For anti-left-invariant pseudo-Euclidean nearly associative algebras, we prove that they are nilpotent of index at most five and characterize them as Jacobi-Jordan-admissible nearly associative algebras. We further show that the associated pseudo-Euclidean Jacobi--Jordan algebras are cyclic. Motivated by the classical double extension of Medina and Revoy, we introduce a double extension procedure for this class of algebras and prove that every anti-left-invariant pseudo-Euclidean nearly associative algebra can be obtained from a trivial pseudo-Euclidean algebra by a finite sequence of such extensions.
For left-invariant pseudo-Euclidean nearly associative algebras, we prove that the associated pseudo-Euclidean Lie algebras are two-step solvable and cyclic. We then develop block, planar, and linear double extensions and show that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra by means of block double extensions. Moreover, we prove that over the field of real numbers, every such algebra can be recursively constructed from a quadratic commutative associative algebra using planar double extensions.
These recursive constructions provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both the anti-left-invariant and left-invariant settings.