Spacecraft attitude: rotation between ECI, LVLH, and Body frames

I'm starting to study spacecraft attitude control and I'm quite confused about the attitude representation in different frames. By reading a bunch of textbooks and papers, I understood that I can simulate a spacecraft's attitude profile by integrating Euler’s rotational equation of motion and the error quaternion kinematics. I've written the equations and I'm able to obtain an attitude profile expressed in the body frame (for example I started by simulating the attitude motion without any external torque and under a constant angular speed along the X axis). So, I end up with a timeseries that contains the spacecraft attitude (expressed through its quaternion and angular velocity with respect to the body frame) at different time instants. I would like to validate my equations of motion by comparing the results with the ones computed with a reliable SW. The problem is that from the SW that I'm using I can only retrieve the quaternion profile described in the inertial (ECI) frame. And here is where my doubts start: how can I rotate my attitude representation in a different frame? I know that I should apply some rotations but I can't figure out which ones. I would like to start from my quaternions expressed in the body frame and obtain the ones referred to the ECI frame. Also, it could be interesting also to understand how to rotate the attitude representation from the body frame to the LVLH frame. I tried to look online but I wasn't able to find an answer to my doubts. Thank you very much for your help!

$\begingroup$ It's much, much more helpful to talk about which frames of reference a rotation connects. What quaternions do you have--ECI to body? And you want ECI to LVLH? Do you already have the state vector (position and velocity) in ECI? $\endgroup$

$\begingroup$ There are a lot of questions and answers about transforming coordinates to LVLH on the site, too. Have you tried searching for LVLH yet? Or the transform tag? $\endgroup$

$\begingroup$ I have the quaternion and the Euler angles ECI to body and I would like to obtain the LVLH to body $\endgroup$

$\begingroup$ One difficulty with LVLH is that different authors define it different ways, as described in space.stackexchange.com/questions/56027/… $\endgroup$

$\begingroup$ This answer shows how to derive an LVLH frame (Ryan C is correct that not all authors use the same conventions, but they'll all be a couple of reflections away from this one) in ECI. Given that, and your ECI to body rotation, you can get LVLH to body. $\endgroup$