BiHom–Lie Brackets and the Toda Equation

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra GL(V) built from two commuting inner automorphisms α=Adψ and β=Adϕ, with ψ,ϕ∈GL(V) and integers i,j. We prove that (GL(V),[·,·](ψ,ϕ)(i,j),α,β) is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice ϕ=ψ with (i,j)=(0,0), the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via L˜=ψ−1Lψ, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded 2×2 blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit 2×2 solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.