Computation of the left invariant vector field for SO(3)

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I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward.

I have been looking at these notes:

https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example
https://math.stackexchange.com/questions/1136592/how-to-calculate-a-vector-in-a-left-invariant-vector-field
https://math.stackexchange.com/questions/1764479/left-invariant-vector-field-on-so3

I have worked out the following for SO(3).

so.jpg


Is this correct? I am not sure about the appearance of the sin and cos after the matrix multiplication.
 

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