In [ Comm. Pure Appl. Math., 28 (1975), pp. 457--478], M. E. Taylor describes a constructive diagonalization method to investigate the reflection of singularities of the solution to first-order hyperbolic systems. According to further works initiated by Engquist and Majda, this approach proved to be well adapted to the construction of artificial boundary conditions. However, in the case of systems governed by pseudodifferential operators with variable coefficients, Taylor's method involves very elaborate calculations of the symbols of the operators. Hence, a direct approach may be difficult when dealing with high-order conditions. This motivates the first part of this paper, where a recursive explicit formulation of Taylor's method is stated for microlocally strictly hyperbolic systems. Consequently, it provides an algorithm leading to symbolic calculations which can be handled by a computer algebra system. In the second part, an application of the method is investigated for the construction of local radiation boundary conditions on arbitrarily shaped surfaces for the transverse electric Maxwell system. It is proved that they are of complete order, provided the introduction of a new symbols class well adapted to the Maxwell system. Next, a complete second-order condition is designed, and when used as an on-surface radiation condition [G. A. Kriegsmann, A. Taflove, and K. R. Umashankar, IEEE Trans. Antennas and Propagation, 35 (1987), pp. 153--161], it can give rise to an ultrafast numerical approximate solution of the scattered field.