

If all you're doing is applying a series of rotations to an object, then start
with the default transformation matrix, and sequentially multiply it by the
rotation matrices of the individual rotations.
If I understand the power of a transformation matrix, then all of the rotations
you perform will be condensed into a single operation so that you can then
perform a rotate <x, y, z> operation with those values.
Easiest way I can think of to work this out is to take a highly asymmetric
object  the Stanford Dragon or just 3 arrowed axes  and instantiate it with
each progressive rotation.
Then in the same scene have the object as transformed by the matrix.
Then you can visually work out the matrix step by step if you have to until it
matches the rotate, rotate, rotate version.
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