The product Jahn-Teller (JT) system with two doubly-degenerate electronic open shells coupling to a single e-mode is studied using an operatorization approach followed by unitary transformation. It is found that the structure of the adiabatic potential energy surface can give anyone of the two different troughs based on thecoupling constants, and the symmetry of the product JT system E ( ×) E can be decomposed into E ( +) A1 ( +) A2.The present approach is conceptually simple and can be extended to other product JT systems.