How to properly change coordinate system using Direction Cosine Matrix?

I am trying to model the behavior of a rocket using Simulink. I use Newton and Euler's laws to compute the accelerations based on the forces and torques on the rocket. I am using two coordinate systems: The flat-Earth coordonate system, that I will denote $E$ ; The rocket frame, $B$ . Let us say that two forces are acting on the rocket: The thruster's force, expressed in $B$ ; The gravity, expressed in $E$ . My goal is to compute the rocket position in $E$ . Using Newton's law, I can compute the acceleration of the rocket in its own frame, $B$ . The gravity force can be passed to the $B$ frame using the Direction Cosine Matrix (DCM), assuming I know the Euler angles. I can integrate the acceleration $\textbf{a}_B$ to find the speed $\textbf{v}_B$ . Then, I can use the DCM to compute $\textbf{v}_E$ , and finally integrate one last time to find $\textbf{x}_E$ . Great, it works, I compared it with Simulink's 6DOF block, and I have the same results. Now, out of curiosity, I tried to change one thing, thinking the result would be the same: I use the DCM before the first integration. Therefore, instead of having: $\textbf{a}_B \xrightarrow{\int} \textbf{v}_B \xrightarrow{DCM} \textbf{v}_E \xrightarrow{\int}\textbf{x}_E$ I have: $\textbf{a}_B \xrightarrow{DCM} \textbf{a}_E \xrightarrow{\int} \textbf{v}_E \xrightarrow{\int}\textbf{x}_E$ And now, the values of $\textbf{x}_E$ are completely different. I'm guessing this is due to the derivative of $\textbf{v}$ in Newton's law that must be expressed in $E$ , but I am unsure why this changes the results. In another project, I had to integrate the raw outputs of the accelerometer of an IMU to find its position in the flat-Earth frame. Using the DCM, I changed the frame of the acceleration first, removed the gravity component, and then integrated twice to find the position. I was therefore doing the frame change before the first integration, and it worked, while for the rocket modelisation I have to do it after. Could you please explain what I am missing ? I suppose I am getting confused with the frames that I am using, but I cannot figure out where. What would be the correct solution for both cases ? Thank you very much for your time, I hope I explained my problem clearly enough.