The computational formula on the constant γ appeared in the equivalently used trace identity and quadratic-form identity

V3 is a 3-dimensional Lie algebra whose commuting operation involves in nine parameters. Appropriately choosing these parameters makes V3 become an arbitrary 3-dimensional Lie algebra. Denote the loop algebra V∼3 generated by V3, from which linear isospectral problems are set up to produce a good many integrable hierarchies of equations. Some of them can be turned into Hamiltonian forms by means of the trace identity, others can be cast into Hamiltonian expressions with the help of the quadratic-form identity. A unified computational formula on the constant γ appeared in the trace identity and the quadratic-form identity is obtained, from which the left problem remarkably proposed in Tu [Tu Guizhang. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys 1989;30(2):330–8] is completely overcome in the paper.