In order to try to explain things and give some examples we can try I thought it might help to show the rotations for a finite subset of the rotation group. We will use the set of rotations of a cube onto itself, this is a permutation group which gives 24 possible rotations as explaned on this page.

heading applied first giving 4 possible orientations:

reference orientation
heading = 0
attitude = 0
bank = 0 |
rotate by 90 degrees about y axis
heading = 90 degrees
attitude = 0
bank = 0 |
rotate by 180 degrees about y axis
heading = 180 degrees
attitude = 0
bank = 0 |
rotate by 270 degrees about y axis
heading = -90 degrees
attitude = 0
bank = 0 |

Then apply attitude +90 degrees for each of the above: (note: that if we went
on to apply bank to these it would just rotate between these values, the straight
up and streight down orientations are known as singularities because they can
be fully defined without using the bank value)

heading = 0
attitude = 90 degrees
bank = 0 |
heading = 90 degrees
attitude = 90 degrees
bank = 0 |
heading = 180 degrees
attitude = 90 degrees
bank = 0 |
heading = -90 degrees
attitude = 90 degrees
bank = 0 |

Or instead apply attitude -90 degrees (also a singularity):

heading = 0
attitude = -90 degrees
bank = 0 |
heading = 90 degrees
attitude = -90 degrees
bank = 0 |
heading = 180 degrees
attitude = -90 degrees
bank = 0 |
heading = -90 degrees
attitude = -90 degrees
bank = 0 |

Normally we dont go beond attitude + or - 90 degrees because thes are singularities,
instead apply bank +90 degrees:

heading = 0
attitude = 0
bank = 90 degrees |
heading = 90 degrees
attitude = 0
bank = 90 degrees |
heading = 180 degrees
attitude = 0
bank = 90 degrees |
heading = -90 degrees
attitude = 0
bank = 90 degrees |

Apply bank +180 degrees:

heading = 0
attitude = 0
bank = 180 degrees |
heading = 90 degrees
attitude = 0
bank = 180 degrees |
heading = 180 degrees
attitude = 0
bank = 180 degrees |
heading = -90 degrees
attitude = 0
bank = 180 degrees |

Apply bank -90 degrees:

heading = 0
attitude = 0
bank = 90 degrees |
heading = 90 degrees
attitude = 0
bank = 90 degrees |
heading = 180 degrees
attitude = 0
bank = 90 degrees |
heading = -90 degrees
attitude = 0
bank = 90 degrees |

encoding of these rotations in quaternions is shown here.

encoding of these rotations in matricies is
shown here.

encoding of these rotations in axis-angle is
shown here.

The working to convert each of these to matrix is shown
here.

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