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In this article the time evolution operator of two interacting quantumoscillators, whose Hamiltonian is an element of the complex $\left\{ h(1)\oplus h(1) \right\} \uplus u(2)$ algebra, is analyzed using the Feynman timeordering operator techniques. This method is consistently used to generate theconditions and to formally find explicit disentangled expressions for suchoperator. In this way, it is shown that all the problem reduces to solve acomplex Riccati-type differential equation. Some closed solutions to thisdifferential equation are found and then concrete disentangling expressions forthe time-ordered evolution operator are given. Finally, the time evolution ofthe coherent states linked to the isotropic 2D quantum oscillator are analyzedunder alternative time-independent an time-dependent Hamiltonian systems.

Time evolution operator for a $\left\{ h(1) \oplus h(1) \right\} \uplus  u(2)$ time-dependent quantum Hamiltonian; a self-consistent resolution method  based on Feynman's disentangling rules