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I have the following Euler's equations (9.23) ($\omega $ is the rigid body's angular velocity, $\Omega$ is the angular velocity of the reference frame whose origin is fixed on the rigid body and whose axes are not fixed on rigid body):

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I don't understand how rewrite it inserting the expression 9.24 for the angular acceleration.

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Adding all three equations of 9.23 gives a matrix form
$$M_{3 \times 1}=I_{3 \times 3} \alpha_{3 \times 1}$$
Here $M$ is the vector torque, $I$ is the moment of inertia tensor and $\alpha$ is the angular acceleration vector. Simplifying $\alpha$ will give the required equations.

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