Studies in the axiomatic formulation of the irreversibility of nuclear fusions with the most general possible SA and NSA forces were initiated in the 1967 Ph. D. Thesis [1] [2] via the generalization of Lie algebras into Albert’s Lie-admissible and Jordan-admissible algebras [3] and the formulation of the first known deformations of Lie algebras with product (Eq. (8) of [1])

which, twenty years later, were followed by tens of thousands of papers on q-deformations. In 1978, after joining the Department of Mathematics of Harvard University under DOE support, Santilli achieved [4] the axiomatic formulation of irreversibility via bi-modules in which motions forward (backward) in time are represented with the ordering of all products to the right (ordering of all products to the left)

The Lie-admissible branch of hadronic mechanics [5] [6] (see also [7]-[17]) is then characterized by two inequivalent iso-mathematics, one per each direction of time, resulting in the Lie-admissible generalization of Heisenberg’s equation for an observable Q in the infinitesimal and finite forms in which SA interactions are represented by the Hamiltonian H and forward NSA interactions are represented by the Santillian S [1] [2]

where e_> (e_<) is the exponentiation in the forward (backward) iso-envelope, with particular case for R = 1 − F/H, S = 1

We then have the time rate of increase of the energy expected for nuclear fusions, showing that Lagrange’s and Hamilton’s external terms F are represented by the Jordan algebra content of the Lie-admissible algebra. Note that given quantum mechanical models of nuclear fusions can be completed into hadronic models via four different simple non-unitary transformations , with a consistent representation of irreversibility, the extended share of nuclear constituents and their broadest possible SA and NSA interactions. Applications to sustainable and controlled nuclear fusions are presented in Refs. [18] [19]. Tutoring lecture [20] are recommendable for physicists.