What is the proper way to obtain the fixed-axis covariance matrix from quaternions?

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REP 103 defines the standard for rotation representation as quaternions, but the covariance matrix associated to it is given in terms of fixed axis rotations. In my work, I use quaternions directly for attitude estimation and obtain covariances in terms of the quaternion parameters.

How can I convert these covariances to the convention used by ROS, so they can be published appropriately?

PS: I have searched ROS Answers and found a few discussions briefly addressing this topic, but usually referring to external documents or additional packages. I think it would be useful to have a single definitive reference for this here, so that implementations don't depend on the reader's interpretation of what is expected.