A note on weakly hyperbolic equations with analytic principal part

In this note we show how to include low order terms in the C^∞$C^{\infty }$ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the C^∞$C^{\infty }$ well-posedness continues to hold.